r"""
Asymptotics of Multivariate Generating Series

Let `F(x) = \sum_{\nu \in \NN^d} F_{\nu} x^\nu` be a multivariate power series
with complex coefficients that converges in a neighborhood of the origin.
Assume that `F = G/H` for some functions `G` and `H` holomorphic in a
neighborhood of the origin. Assume also that `H` is a polynomial.

This computes asymptotics for the coefficients `F_{r \alpha}` as `r \to \infty`
with `r \alpha \in \NN^d` for `\alpha` in a permissible subset of `d`-tuples of
positive reals. More specifically, it computes arbitrary terms of the
asymptotic expansion for `F_{r \alpha}` when the asymptotics are controlled by
a strictly minimal multiple point of the algebraic variety `H = 0`.

The algorithms and formulas implemented here come from [RaWi2008a]_
and [RaWi2012]_. For a general reference take a look in the book [PeWi2013].

.. [AiYu1983] \I.A. Aizenberg and A.P. Yuzhakov.
   *Integral representations and residues in multidimensional complex analysis*.
   Translations of Mathematical Monographs, **58**. American Mathematical
   Society, Providence, RI. (1983). x+283 pp. ISBN: 0-8218-4511-X.

.. [Raic2012] Alexander Raichev.
   *Leinartas's partial fraction decomposition*.
   :arxiv:`1206.4740`.

.. [RaWi2008a] Alexander Raichev and Mark C. Wilson. *Asymptotics of
   coefficients of multivariate generating functions: improvements for
   smooth points*, Electronic Journal of Combinatorics, Vol. 15 (2008).
   R89 :arxiv:`0803.2914`.

.. [RaWi2012] Alexander Raichev and Mark C. Wilson. *Asymptotics of
   coefficients of multivariate generating functions: improvements for
   smooth points*. Online Journal of Analytic Combinatorics.
   Issue 6, (2011). :arxiv:`1009.5715`.

.. [PeWi2013] Robin Pemantle and Mark C. Wilson.
   *Analytic Combinatorics in Several Variables*.
   Cambridge University Press, 2013.


Introductory Examples
=====================

::

    sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing

A univariate smooth point example::

    sage: R.<x> = PolynomialRing(QQ)
    sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
    sage: H = (x - 1/2)^3
    sage: Hfac = H.factor()
    sage: G = -1/(x + 3)/Hfac.unit()
    sage: F = FFPD(G, Hfac)
    sage: F
    (-1/(x + 3), [(x - 1/2, 3)])
    sage: alpha = [1]
    sage: decomp = F.asymptotic_decomposition(alpha)
    sage: decomp
    (0, []) +
    (-1/2*(x^2 + 6*x + 9)*r^2/(x^5 + 9*x^4 + 27*x^3 + 27*x^2)
     - 1/2*(5*x^2 + 24*x + 27)*r/(x^5 + 9*x^4 + 27*x^3 + 27*x^2)
     - 3*(x^2 + 3*x + 3)/(x^5 + 9*x^4 + 27*x^3 + 27*x^2),
     [(x - 1/2, 1)])
    sage: F1 = decomp[1]
    sage: p = {x: 1/2}
    sage: asy = F1.asymptotics(p, alpha, 3)
    sage: asy
    (8/343*(49*r^2 + 161*r + 114)*2^r, 2, 8/7*r^2 + 184/49*r + 912/343)
    sage: F.relative_error(asy[0], alpha, [1, 2, 4, 8, 16], asy[1])
    [((1,), 7.555555556, [7.556851312], [-0.0001714971672]),
     ((2,), 14.74074074, [14.74052478], [0.00001465051901]),
     ((4,), 35.96502058, [35.96501458], [1.667911934e-7]),
     ((8,), 105.8425656, [105.8425656], [4.399565380e-11]),
     ((16,), 355.3119534, [355.3119534], [0.0000000000])]

Another smooth point example (Example 5.4 of [RaWi2008a]_)::

    sage: R.<x,y> = PolynomialRing(QQ)
    sage: FFPD = FractionWithFactoredDenominatorRing(R)
    sage: q = 1/2
    sage: qq = q.denominator()
    sage: H = 1 - q*x + q*x*y - x^2*y
    sage: Hfac = H.factor()
    sage: G = (1 - q*x)/Hfac.unit()
    sage: F = FFPD(G, Hfac)
    sage: alpha = list(qq*vector([2, 1 - q]))
    sage: alpha
    [4, 1]
    sage: I = F.smooth_critical_ideal(alpha)
    sage: I
    Ideal (y^2 - 2*y + 1, x + 1/4*y - 5/4) of
     Multivariate Polynomial Ring in x, y over Rational Field
    sage: s = solve([SR(z) for z in I.gens()],
    ....:           [SR(z) for z in R.gens()], solution_dict=true)
    sage: s == [{SR(x): 1, SR(y): 1}]
    True
    sage: p = s[0]
    sage: asy = F.asymptotics(p, alpha, 1, verbose=True)
    Creating auxiliary functions...
    Computing derivatives of auxiliary functions...
    Computing derivatives of more auxiliary functions...
    Computing second order differential operator actions...
    sage: asy
    (1/24*2^(2/3)*(sqrt(3) + 4/(sqrt(3) + I) + I)*gamma(1/3)/(pi*r^(1/3)),
     1,
     1/24*2^(2/3)*(sqrt(3) + 4/(sqrt(3) + I) + I)*gamma(1/3)/(pi*r^(1/3)))
    sage: r = SR('r')
    sage: tuple((a*r^(1/3)).full_simplify() / r^(1/3) for a in asy)  # make nicer coefficients
    (1/12*sqrt(3)*2^(2/3)*gamma(1/3)/(pi*r^(1/3)),
     1,
     1/12*sqrt(3)*2^(2/3)*gamma(1/3)/(pi*r^(1/3)))
    sage: F.relative_error(asy[0], alpha, [1, 2, 4, 8, 16], asy[1])
    [((4, 1), 0.1875000000, [0.1953794675...], [-0.042023826...]),
     ((8, 2), 0.1523437500, [0.1550727862...], [-0.017913673...]),
     ((16, 4), 0.1221771240, [0.1230813519...], [-0.0074009592...]),
     ((32, 8), 0.09739671811, [0.09768973377...], [-0.0030084757...]),
     ((64, 16), 0.07744253816, [0.07753639308...], [-0.0012119297...])]

A multiple point example (Example 6.5 of [RaWi2012]_)::

    sage: R.<x,y> = PolynomialRing(QQ)
    sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
    sage: H = (1 - 2*x - y)**2 * (1 - x - 2*y)**2
    sage: Hfac = H.factor()
    sage: G = 1/Hfac.unit()
    sage: F = FFPD(G, Hfac)
    sage: F
    (1, [(x + 2*y - 1, 2), (2*x + y - 1, 2)])
    sage: I = F.singular_ideal()
    sage: I
    Ideal (x - 1/3, y - 1/3) of
     Multivariate Polynomial Ring in x, y over Rational Field
    sage: p = {x: 1/3, y: 1/3}
    sage: F.is_convenient_multiple_point(p)
    (True, 'convenient in variables [x, y]')
    sage: alpha = (var('a'), var('b'))
    sage: decomp =  F.asymptotic_decomposition(alpha); decomp
    (0, []) +
    (-1/9*(2*b^2*x^2 - 5*a*b*x*y + 2*a^2*y^2)*r^2/(x^2*y^2)
      - 1/9*(6*b*x^2 - 5*(a + b)*x*y + 6*a*y^2)*r/(x^2*y^2)
      - 1/9*(4*x^2 - 5*x*y + 4*y^2)/(x^2*y^2),
     [(x + 2*y - 1, 1), (2*x + y - 1, 1)])
    sage: F1 = decomp[1]
    sage: F1.asymptotics(p, alpha, 2)
    (-3*((2*a^2 - 5*a*b + 2*b^2)*r^2 + (a + b)*r + 3)*((1/3)^(-a)*(1/3)^(-b))^r,
     (1/3)^(-a)*(1/3)^(-b), -3*(2*a^2 - 5*a*b + 2*b^2)*r^2 - 3*(a + b)*r - 9)
    sage: alpha = [4, 3]
    sage: decomp =  F.asymptotic_decomposition(alpha)
    sage: F1 = decomp[1]
    sage: asy = F1.asymptotics(p, alpha, 2)
    sage: asy
    (3*(10*r^2 - 7*r - 3)*2187^r, 2187, 30*r^2 - 21*r - 9)
    sage: F.relative_error(asy[0], alpha, [1, 2, 4, 8], asy[1])
    [((4, 3), 30.72702332, [0.0000000000], [1.000000000]),
     ((8, 6), 111.9315678, [69.00000000], [0.3835519207]),
     ((16, 12), 442.7813138, [387.0000000], [0.1259793763]),
     ((32, 24), 1799.879232, [1743.000000], [0.03160169385])]

TESTS::

    sage: R.<x,y> = PolynomialRing(QQ)
    sage: FFPD = FractionWithFactoredDenominatorRing(R)
    sage: H = (1 - 2*x - y) * (1 - x - 2*y)
    sage: G = 1
    sage: Hfac = H.factor()
    sage: G = G / Hfac.unit()
    sage: F = FFPD(G, Hfac); F
    (1, [(x + 2*y - 1, 1), (2*x + y - 1, 1)])
    sage: p = {x: 1, y: 1}
    sage: alpha = [1, 1]
    sage: F.asymptotics(p, alpha, 1)
    (1/3, 1, 1/3)

::

    sage: R.<x,y,t> = PolynomialRing(QQ)
    sage: FFPD = FractionWithFactoredDenominatorRing(R)
    sage: H = (1 - y) * (1 + x^2) * (1 - t*(1 + x^2 + x*y^2))
    sage: G = (1 + x) * (1 + x^2 - x*y^2)
    sage: Hfac = H.factor()
    sage: G = G / Hfac.unit()
    sage: F = FFPD(G, Hfac); F
    (-x^2*y^2 + x^3 - x*y^2 + x^2 + x + 1,
     [(y - 1, 1), (x^2 + 1, 1), (x*y^2*t + x^2*t + t - 1, 1)])
    sage: p = {x: 1, y: 1, t: 1/3}
    sage: alpha = [1, 1, 1]
    sage: F.asymptotics_multiple(p, alpha, 1, var('r'))  # not tested - see #19989


Various
=======

AUTHORS:

- Alexander Raichev (2008)
- Daniel Krenn (2014, 2016)


Classes and Methods
===================
"""
#*****************************************************************************
#       Copyright (C) 2008 Alexander Raichev <tortoise.said@gmail.com>
#       Copyright (C) 2014, 2016 Daniel Krenn <dev@danielkrenn.at>
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
#                  http://www.gnu.org/licenses/
#*****************************************************************************
from __future__ import print_function
from six.moves import range

from functools import total_ordering
from itertools import combinations_with_replacement
from sage.structure.element import RingElement
from sage.structure.unique_representation import UniqueRepresentation
from sage.rings.ring import Ring
from sage.calculus.var import var
from sage.calculus.functional import diff
from sage.symbolic.ring import SR
from sage.misc.misc_c import prod
from sage.rings.integer import Integer
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.categories.rings import Rings
from sage.structure.richcmp import richcmp_by_eq_and_lt


@total_ordering
class FractionWithFactoredDenominator(RingElement):
    r"""
    This element represents a fraction with a factored polynomial
    denominator. See also its parent
    :class:`FractionWithFactoredDenominatorRing` for details.

    Represents a fraction with factored polynomial denominator (FFPD)
    `p/(q_1^{e_1} \cdots q_n^{e_n})` by storing the parts `p` and
    `[(q_1, e_1), \ldots, (q_n, e_n)]`.
    Here `q_1, \ldots, q_n` are elements of a 0- or multi-variate factorial
    polynomial ring `R` , `q_1, \ldots, q_n` are distinct irreducible elements
    of `R` , `e_1, \ldots, e_n` are positive integers, and `p` is a function
    of the indeterminates of `R` (e.g., a Sage symbolic expression). An
    element `r` with no polynomial denominator is represented as ``(r, [])``.

    INPUT:

    - ``numerator`` -- an element `p`; this can be of any ring from which
      parent's base has coercion in
    - ``denominator_factored`` -- a list of the form
      `[(q_1, e_1), \ldots, (q_n, e_n)]`, where the `q_1, \ldots, q_n` are
      distinct irreducible elements of `R` and the `e_i` are positive
      integers
    - ``reduce`` -- (optional) if ``True``, then represent
      `p/(q_1^{e_1} \cdots q_n^{e_n})` in lowest terms, otherwise
      this won't attempt to divide `p` by any of the `q_i`

    OUTPUT:

    An element representing the rational expression
    `p/(q_1^{e_1} \cdots q_n^{e_n})`.

    EXAMPLES::

        sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
        sage: R.<x,y> = PolynomialRing(QQ)
        sage: FFPD = FractionWithFactoredDenominatorRing(R)
        sage: df = [x, 1], [y, 1], [x*y+1, 1]
        sage: f = FFPD(x, df)
        sage: f
        (1, [(y, 1), (x*y + 1, 1)])
        sage: ff = FFPD(x, df, reduce=False)
        sage: ff
        (x, [(y, 1), (x, 1), (x*y + 1, 1)])

        sage: f = FFPD(x + y, [(x + y, 1)])
        sage: f
        (1, [])

    ::

        sage: R.<x> = PolynomialRing(QQ)
        sage: FFPD = FractionWithFactoredDenominatorRing(R)
        sage: f = 5*x^3 + 1/x + 1/(x-1) + 1/(3*x^2 + 1)
        sage: FFPD(f)
        (5*x^7 - 5*x^6 + 5/3*x^5 - 5/3*x^4 + 2*x^3 - 2/3*x^2 + 1/3*x - 1/3,
        [(x - 1, 1), (x, 1), (x^2 + 1/3, 1)])

    ::

        sage: R.<x,y> = PolynomialRing(QQ)
        sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
        sage: f = 2*y/(5*(x^3 - 1)*(y + 1))
        sage: FFPD(f)
        (2/5*y, [(y + 1, 1), (x - 1, 1), (x^2 + x + 1, 1)])

        sage: p = 1/x^2
        sage: q = 3*x**2*y
        sage: qs = q.factor()
        sage: f = FFPD(p/qs.unit(), qs)
        sage: f
        (1/3/x^2, [(y, 1), (x, 2)])

        sage: f = FFPD(cos(x)*x*y^2, [(x, 2), (y, 1)])
        sage: f
        (x*y^2*cos(x), [(y, 1), (x, 2)])

        sage: G = exp(x + y)
        sage: H = (1 - 2*x - y) * (1 - x - 2*y)
        sage: a = FFPD(G/H)
        sage: a
        (e^(x + y), [(x + 2*y - 1, 1), (2*x + y - 1, 1)])
        sage: a.denominator_ring
        Multivariate Polynomial Ring in x, y over Rational Field
        sage: b = FFPD(G, H.factor())
        sage: b
        (e^(x + y), [(x + 2*y - 1, 1), (2*x + y - 1, 1)])
        sage: b.denominator_ring
        Multivariate Polynomial Ring in x, y over Rational Field

    Singular throws a 'not implemented' error when trying to factor in
    a multivariate polynomial ring over an inexact field::

        sage: R.<x,y> = PolynomialRing(CC)
        sage: FFPD = FractionWithFactoredDenominatorRing(R)
        sage: f = (x + 1)/(x*y*(x*y + 1)^2)
        sage: FFPD(f)
        Traceback (most recent call last):
        ...
        TypeError: Singular error:
           ? not implemented
           ? error occurred in or before STDIN line ...:
           `def sage...=factorize(sage...);`

    AUTHORS:

    - Alexander Raichev (2012-07-26)
    - Daniel Krenn (2014-12-01)
    """

    def __init__(self, parent, numerator, denominator_factored, reduce=True):
        r"""
        Initialize ``self``.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: df = [x, 1], [y, 1], [x*y+1, 1]
            sage: f = FFPD(x, df)
            sage: TestSuite(f).run()
        """
        super(FractionWithFactoredDenominator, self).__init__(parent)

        from sage.rings.semirings.non_negative_integer_semiring import NN
        self._numerator = parent._numerator_ring(numerator)
        self._denominator_factored = list(
            (parent._denominator_ring(d), NN(n))
            for d, n in denominator_factored)

        R = self.denominator_ring
        if numerator in R and reduce:
            # Reduce fraction if possible.
            numer = R(self._numerator)
            df = self._denominator_factored
            new_df = []
            for (q, e) in df:
                ee = e
                quo, rem = numer.quo_rem(q)
                while rem == 0 and ee > 0:
                    ee -= 1
                    numer = quo
                    quo, rem = numer.quo_rem(q)
                if ee > 0:
                    new_df.append((q, ee))
            self._numerator = numer
            self._denominator_factored = new_df

    def numerator(self):
        r"""
        Return the numerator of ``self``.

        OUTPUT:

        The numerator.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: H = (1 - x - y - x*y)**2*(1-x)
            sage: Hfac = H.factor()
            sage: G = exp(y)/Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: F.numerator()
            -e^y
        """
        return self._numerator

    def denominator(self):
        r"""
        Return the denominator of ``self``.

        OUTPUT:

        The denominator (i.e., the product of the factored denominator).

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: H = (1 - x - y - x*y)**2*(1-x)
            sage: Hfac = H.factor()
            sage: G = exp(y)/Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: F.denominator()
            x^3*y^2 + 2*x^3*y + x^2*y^2 + x^3 - 2*x^2*y - x*y^2 - 3*x^2 - 2*x*y
            - y^2 + 3*x + 2*y - 1
        """
        return prod(q ** e for q, e in self.denominator_factored())

    def denominator_factored(self):
        r"""
        Return the factorization in ``self.denominator_ring`` of the denominator of
        ``self`` but without the unit part.

        OUTPUT:

        The factored denominator as a list of tuple ``(f, m)``, where `f` is
        a factor and `m` its multiplicity.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: H = (1 - x - y - x*y)**2*(1-x)
            sage: Hfac = H.factor()
            sage: G = exp(y)/Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: F.denominator_factored()
            [(x - 1, 1), (x*y + x + y - 1, 2)]
        """
        return self._denominator_factored

    @property
    def denominator_ring(self):
        r"""
        Return the ring of the denominator.

        OUTPUT:

        A ring.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: H = (1 - x - y - x*y)**2*(1-x)
            sage: Hfac = H.factor()
            sage: G = exp(y)/Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: F.denominator_ring
            Multivariate Polynomial Ring in x, y over Rational Field
            sage: F = FFPD(G/H)
            sage: F
            (e^y, [(x - 1, 1), (x*y + x + y - 1, 2)])
            sage: F.denominator_ring
            Multivariate Polynomial Ring in x, y over Rational Field
        """
        return self.parent()._denominator_ring

    @property
    def numerator_ring(self):
        r"""
        Return the ring of the numerator.

        OUTPUT:

        A ring.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: H = (1 - x - y - x*y)**2*(1-x)
            sage: Hfac = H.factor()
            sage: G = exp(y)/Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: F.numerator_ring
            Symbolic Ring
            sage: F = FFPD(G/H)
            sage: F
            (e^y, [(x - 1, 1), (x*y + x + y - 1, 2)])
            sage: F.numerator_ring
            Symbolic Ring
        """
        return self.parent()._numerator_ring

    def dimension(self):
        r"""
        Return the number of indeterminates of ``self.denominator_ring``.

        OUTPUT:

        An integer.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: H = (1 - x - y - x*y)**2*(1-x)
            sage: Hfac = H.factor()
            sage: G = exp(y)/Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: F.dimension()
            2
        """
        from sage.rings.polynomial.polynomial_ring import is_PolynomialRing
        from sage.rings.polynomial.multi_polynomial_ring_generic import is_MPolynomialRing
        R = self.denominator_ring
        if is_PolynomialRing(R) or is_MPolynomialRing(R):
            return R.ngens()
        raise NotImplementedError('only polynomial rings are supported as base')

    def quotient(self):
        r"""
        Convert ``self`` into a quotient.

        OUTPUT:

        An element.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: H = (1 - x - y - x*y)**2*(1-x)
            sage: Hfac = H.factor()
            sage: G = exp(y)/Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: F
            (-e^y, [(x - 1, 1), (x*y + x + y - 1, 2)])
            sage: F.quotient()
            -e^y/(x^3*y^2 + 2*x^3*y + x^2*y^2 + x^3 - 2*x^2*y - x*y^2 - 3*x^2 -
            2*x*y - y^2 + 3*x + 2*y - 1)
        """
        return self.numerator() / self.denominator()

    def _repr_(self):
        r"""
        Return a string representation of ``self``.

        OUTPUT:

        A string.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: f = FFPD(x + y, [(y, 1), (x, 1)])
            sage: f
            (x + y, [(y, 1), (x, 1)])
        """
        return repr((self.numerator(), self.denominator_factored()))

    _richcmp_ = richcmp_by_eq_and_lt("_eq_", "_lt_")

    def _eq_(self, other):
        r"""
        Return whether the FFPD instance ``other`` is equal to
        this FFPD instance.

        Two FFPD instances are equal iff they represent the same
        fraction.

        INPUT:

        - ``other`` -- an instance of :class:`FractionWithFactoredDenominator`

        OUTPUT:

        ``True`` or ``False``.

        It can be assumed that ``self`` and ``other`` have the same parent.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: df = [x, 1], [y, 1], [x*y+1, 1]
            sage: f = FFPD(x, df)
            sage: ff = FFPD(x, df, reduce=False)
            sage: f == ff
            True
            sage: g = FFPD(y, df)
            sage: g == f
            False

        ::

            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: G = exp(x + y)
            sage: H = (1 - 2*x - y) * (1 - x - 2*y)
            sage: a = FFPD(G/H)
            sage: b = FFPD(G, H.factor())
            sage: bool(a == b)
            True

        ::

            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: df = [x, 1], [y, 1], [x*y+1, 1]
            sage: f = FFPD(x, df)
            sage: ff = FFPD(x, df, reduce=False)
            sage: f != ff
            False
            sage: g = FFPD(y, df)
            sage: g != f
            True

        TESTS::

            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: f = FFPD(x, [])
            sage: f == x
            True

        ::

            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: f = FFPD(x*y, [(x-1, 1), (y-2, 2)])
            sage: g = FFPD(x, [(x-1, 1), (y-2, 2)])
            sage: f == f
            True
            sage: f == g
            False

            sage: f < g
            Traceback (most recent call last):
            ...
            AttributeError:
            'FractionWithFactoredDenominatorRing_with_category.element_class'
            object has no attribute '_lt_'
        """
        return (self.numerator() * other.denominator() ==
                other.numerator() * self.denominator())

    def _total_order_key_(self):
        r"""
        Return a key that can be used for sorting.

        FFPD ``A`` is less than FFPD ``B`` iff
        (the denominator factorization of ``A`` is shorter than that of ``B``)
        of (the denominator factorization lengths are equal and
        the denominator of ``A`` is less than that of ``B`` in their ring) or
        (the denominator factorization lengths are equal and the
        denominators are equal and the numerator of ``A`` is less than that
        of ``B`` in their ring).

        OUTPUT:

        A tuple.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: df = [x, 1], [y, 1], [x*y+1, 1]
            sage: f = FFPD(x, df); f
            (1, [(y, 1), (x*y + 1, 1)])
            sage: ff = FFPD(x, df, reduce=False); ff
            (x, [(y, 1), (x, 1), (x*y + 1, 1)])
            sage: g = FFPD(y, df)
            sage: h = FFPD(exp(x), df)
            sage: i = FFPD(sin(x + 2), df)
            sage: f._total_order_key_() < ff._total_order_key_()
            True
            sage: f._total_order_key_() < g._total_order_key_()
            True
            sage: g._total_order_key_() < h._total_order_key_()
            True
            sage: bool(h._total_order_key_() < i._total_order_key_())
            False
        """
        return (len(self.denominator_factored()),
                self.denominator(),
                self.numerator())

    def univariate_decomposition(self):
        r"""
        Return the usual univariate partial fraction decomposition
        of ``self``.

        Assume that the numerator of ``self`` lies in the same univariate
        factorial polynomial ring as the factors of the denominator.

        Let `f = p/q` be a rational expression where `p` and `q` lie in a
        univariate factorial polynomial ring `R`.
        Let `q_1^{e_1} \cdots q_n^{e_n}` be the
        unique factorization of `q` in `R` into irreducible factors.
        Then `f` can be written uniquely as:

        .. MATH::

            (*) \quad p_0 + \sum_{i=1}^{m} \frac{p_i}{q_i^{e_i}},

        for some `p_j \in R`.
        We call `(*)` the *usual partial fraction decomposition* of `f`.

        .. NOTE::

            This partial fraction decomposition can be computed using
            :meth:`~sage.symbolic.expression.Expression.partial_fraction` or
            :meth:`~sage.categories.quotient_fields.QuotientFields.ElementMethods.partial_fraction_decomposition`
            as well. However, here we use the already obtained/cached
            factorization of the denominator. This gives a speed up for
            non-small instances.

        OUTPUT:

        An instance of :class:`FractionWithFactoredDenominatorSum`.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing

        One variable::

            sage: R.<x> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: f = 5*x^3 + 1/x + 1/(x-1) + 1/(3*x^2 + 1)
            sage: f
            (15*x^7 - 15*x^6 + 5*x^5 - 5*x^4 + 6*x^3 - 2*x^2 + x - 1)/(3*x^4 -
            3*x^3 + x^2 - x)
            sage: decomp = FFPD(f).univariate_decomposition()
            sage: decomp
            (5*x^3, []) +
            (1, [(x - 1, 1)]) +
            (1, [(x, 1)]) +
            (1/3, [(x^2 + 1/3, 1)])
            sage: decomp.sum().quotient() == f
            True

        One variable with numerator in symbolic ring::

            sage: R.<x> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: f = 5*x^3 + 1/x + 1/(x-1) + exp(x)/(3*x^2 + 1)
            sage: f
            (5*x^5 - 5*x^4 + 2*x - 1)/(x^2 - x) + e^x/(3*x^2 + 1)
            sage: decomp = FFPD(f).univariate_decomposition()
            sage: decomp
            (0, []) +
            (15/4*x^7 - 15/4*x^6 + 5/4*x^5 - 5/4*x^4 + 3/2*x^3 + 1/4*x^2*e^x -
             3/4*x^2 - 1/4*x*e^x + 1/2*x - 1/4, [(x - 1, 1)]) +
            (-15*x^7 + 15*x^6 - 5*x^5 + 5*x^4 - 6*x^3 -
             x^2*e^x + 3*x^2 + x*e^x - 2*x + 1, [(x, 1)]) +
            (1/4*(15*x^7 - 15*x^6 + 5*x^5 - 5*x^4 + 6*x^3 + x^2*e^x -
                  3*x^2 - x*e^x + 2*x - 1)*(3*x - 1), [(x^2 + 1/3, 1)])

        One variable over a finite field::

            sage: R.<x> = PolynomialRing(GF(2))
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: f = 5*x^3 + 1/x + 1/(x-1) + 1/(3*x^2 + 1)
            sage: f
            (x^6 + x^4 + 1)/(x^3 + x)
            sage: decomp = FFPD(f).univariate_decomposition()
            sage: decomp
            (x^3, []) + (1, [(x, 1)]) + (x, [(x + 1, 2)])
            sage: decomp.sum().quotient() == f
            True

        One variable over an inexact field::

            sage: R.<x> = PolynomialRing(CC)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: f = 5*x^3 + 1/x + 1/(x-1) + 1/(3*x^2 + 1)
            sage: f
            (15.0000000000000*x^7 - 15.0000000000000*x^6 + 5.00000000000000*x^5
             - 5.00000000000000*x^4 + 6.00000000000000*x^3
             - 2.00000000000000*x^2 + x - 1.00000000000000)/(3.00000000000000*x^4
             - 3.00000000000000*x^3 + x^2 - x)
            sage: decomp = FFPD(f).univariate_decomposition()
            sage: decomp
            (5.00000000000000*x^3, []) +
            (1.00000000000000, [(x - 1.00000000000000, 1)]) +
            (-0.288675134594813*I, [(x - 0.577350269189626*I, 1)]) +
            (1.00000000000000, [(x, 1)]) +
            (0.288675134594813*I, [(x + 0.577350269189626*I, 1)])
            sage: decomp.sum().quotient() == f # Rounding error coming
            False

        TESTS::

            sage: R.<x> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: f = exp(x) / (x^2-x)
            sage: f
            e^x/(x^2 - x)
            sage: FFPD(f).univariate_decomposition()
            (0, []) + (e^x, [(x - 1, 1)]) + (-e^x, [(x, 1)])

        AUTHORS:

        - Robert Bradshaw (2007-05-31)
        - Alexander Raichev (2012-06-25)
        - Daniel Krenn (2014-12-01)
        """
        if self.dimension() > 1:
            return FractionWithFactoredDenominatorSum([self])

        R = self.denominator_ring
        p = self.numerator()
        q = self.denominator()
        try:
            whole, p = R(p).quo_rem(q)
            mn = R.one()
        except (TypeError, ValueError):
            whole = R(0)
            mn = p
            p = R.one()
        df = self.denominator_factored()
        decomp = [self.parent()(whole, [])]
        denominator = prod(b**n for b, n in df)
        for a, m in df:
            am = a**m
            q, r = denominator.quo_rem(am)
            assert r==0
            numer = p * q.inverse_mod(am) % am
            # The inverse exists because the product and a**m
            # are relatively prime.
            decomp.append(self.parent()(mn * numer, [(a, m)]))
        return FractionWithFactoredDenominatorSum(decomp)

    def nullstellensatz_certificate(self):
        r"""
        Return a Nullstellensatz certificate of ``self`` if it exists.

        Let `[(q_1, e_1), \ldots, (q_n, e_n)]` be the denominator
        factorization of ``self``. The Nullstellensatz certificate is
        a list of polynomials `h_1, \ldots, h_m` in ``self.denominator_ring``
        that satisfies `h_1 q_1 + \cdots + h_m q_n = 1` if it exists.

        .. NOTE::

            Only works for multivariate base rings.

        OUTPUT:

        A list of polynomials or ``None`` if no Nullstellensatz
        certificate exists.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: G = sin(x)
            sage: H = x^2 * (x*y + 1)
            sage: f = FFPD(G, H.factor())
            sage: L = f.nullstellensatz_certificate()
            sage: L
            [y^2, -x*y + 1]
            sage: df = f.denominator_factored()
            sage: sum(L[i]*df[i][0]**df[i][1] for i in range(len(df))) == 1
            True

        ::

            sage: f = 1/(x*y)
            sage: L = FFPD(f).nullstellensatz_certificate()
            sage: L is None
            True
        """
        R = self.denominator_ring
        df = self.denominator_factored()
        J = R.ideal([q ** e for q, e in df])
        if R.one() in J:
            return R.one().lift(J)
        return None

    def nullstellensatz_decomposition(self):
        r"""
        Return a Nullstellensatz decomposition of ``self``.

        Let `f = p/q` where `q` lies in a `d` -variate polynomial ring
        `K[X]` for some field `K` and `d \geq 1`.
        Let `q_1^{e_1} \cdots q_n^{e_n}` be the
        unique factorization of `q` in `K[X]` into irreducible factors and
        let `V_i` be the algebraic variety `\{x \in L^d \mid q_i(x) = 0\}`
        of `q_i` over the algebraic closure `L` of `K`.
        By [Raic2012]_, `f` can be written as

        .. MATH::

            (*) \quad \sum_A \frac{p_A}{\prod_{i \in A} q_i^{e_i}},

        where the `p_A` are products of `p` and elements in `K[X]` and
        the sum is taken over all subsets
        `A \subseteq \{1, \ldots, m\}` such that
        `\bigcap_{i\in A} T_i \neq \emptyset`.

        We call `(*)` a *Nullstellensatz decomposition* of `f`.
        Nullstellensatz decompositions are not unique.

        The algorithm used comes from [Raic2012]_.

        .. NOTE::

            Recursive. Only works for multivariate ``self``.

        OUTPUT:

        An instance of :class:`FractionWithFactoredDenominatorSum`.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import *
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: f = 1/(x*(x*y + 1))
            sage: decomp = FFPD(f).nullstellensatz_decomposition()
            sage: decomp
            (0, []) + (1, [(x, 1)]) + (-y, [(x*y + 1, 1)])
            sage: decomp.sum().quotient() == f
            True
            sage: [r.nullstellensatz_certificate() is None for r in decomp]
            [True, True, True]

        ::

            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: G = sin(y)
            sage: H = x*(x*y + 1)
            sage: f = FFPD(G, H.factor())
            sage: decomp = f.nullstellensatz_decomposition()
            sage: decomp
            (0, []) + (sin(y), [(x, 1)]) + (-y*sin(y), [(x*y + 1, 1)])
            sage: bool(decomp.sum().quotient() == G/H)
            True
            sage: [r.nullstellensatz_certificate() is None for r in decomp]
            [True, True, True]
        """
        L = self.nullstellensatz_certificate()
        if L is None:
            # No decomposing possible.
            return FractionWithFactoredDenominatorSum([self])

        # Otherwise decompose recursively.
        decomp = FractionWithFactoredDenominatorSum()
        p = self.numerator()
        df = self.denominator_factored()
        m = len(df)
        iteration1 = FractionWithFactoredDenominatorSum(
            [self.parent()(p * L[i], [df[j] for j in range(m) if j != i])
             for i in range(m) if L[i] != 0])

        # Now decompose each FFPD of iteration1.
        for r in iteration1:
            decomp.extend(r.nullstellensatz_decomposition())

        # Simplify and return result.
        return decomp._combine_like_terms_().whole_and_parts()

    def algebraic_dependence_certificate(self):
        r"""
        Return the algebraic dependence certificate of ``self``.

        The algebraic dependence certificate is the ideal `J` of
        annihilating polynomials for the set of polynomials
        ``[q^e for (q, e) in self.denominator_factored()]``,
        which could be the zero ideal.
        The ideal `J` lies in a polynomial ring over the field
        ``self.denominator_ring.base_ring()`` that has
        ``m = len(self.denominator_factored())`` indeterminates.

        OUTPUT:

        An ideal.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: f = 1/(x^2 * (x*y + 1) * y^3)
            sage: ff = FFPD(f)
            sage: J = ff.algebraic_dependence_certificate(); J
            Ideal (1 - 6*T2 + 15*T2^2 - 20*T2^3 + 15*T2^4 - T0^2*T1^3 -
             6*T2^5  + T2^6) of Multivariate Polynomial Ring in
             T0, T1, T2 over Rational Field
            sage: g = J.gens()[0]
            sage: df = ff.denominator_factored()
            sage: g(*(q**e for q, e in df)) == 0
            True

        ::

            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: G = exp(x + y)
            sage: H = x^2 * (x*y + 1) * y^3
            sage: ff = FFPD(G, H.factor())
            sage: J = ff.algebraic_dependence_certificate(); J
            Ideal (1 - 6*T2 + 15*T2^2 - 20*T2^3 + 15*T2^4 - T0^2*T1^3 -
            6*T2^5 + T2^6) of Multivariate Polynomial Ring in
            T0, T1, T2 over Rational Field
            sage: g = J.gens()[0]
            sage: df = ff.denominator_factored()
            sage: g(*(q**e for q, e in df)) == 0
            True

        ::

            sage: f = 1/(x^3 * y^2)
            sage: J = FFPD(f).algebraic_dependence_certificate()
            sage: J
            Ideal (0) of Multivariate Polynomial Ring in T0, T1 over Rational Field

        ::

            sage: f = sin(1)/(x^3 * y^2)
            sage: J = FFPD(f).algebraic_dependence_certificate()
            sage: J
            Ideal (0) of Multivariate Polynomial Ring in T0, T1 over Rational Field
        """
        R = self.denominator_ring
        df = self.denominator_factored()
        if not df:
            return R.ideal()    # The zero ideal.
        m = len(df)
        F = R.base_ring()
        Xs = list(R.gens())
        d = len(Xs)

        # Expand R by 2 * m new variables.
        S = 'S'
        while S in [str(x) for x in Xs]:
            S = S + 'S'
        Ss = [S + str(i) for i in range(m)]
        T = 'T'
        while T in [str(x) for x in Xs]:
            T = T + 'T'
        Ts = [T + str(i) for i in range(m)]

        Vs = [str(x) for x in Xs] + Ss + Ts
        RR = PolynomialRing(F, Vs)
        Xs = RR.gens()[:d]
        Ss = RR.gens()[d: d + m]
        Ts = RR.gens()[d + m: d + 2 * m]

        # Compute the appropriate elimination ideal.
        J = RR.ideal([Ss[j] - RR(df[j][0]) for j in range(m)] +
                     [Ss[j] ** df[j][1] - Ts[j] for j in range(m)])
        J = J.elimination_ideal(Xs + Ss)

        # Coerce J into the polynomial ring in the indeteminates Ts[m:].
        # I choose the negdeglex order because i find it useful in my work.
        RRR = PolynomialRing(F, [str(t) for t in Ts], order='negdeglex')
        return RRR.ideal(J)

    def algebraic_dependence_decomposition(self, whole_and_parts=True):
        r"""
        Return an algebraic dependence decomposition of ``self``.

        Let `f = p/q` where `q` lies in a `d`-variate polynomial ring
        `K[X]` for some field `K`.
        Let `q_1^{e_1} \cdots q_n^{e_n}` be the
        unique factorization of `q` in `K[X]` into irreducible factors and
        let `V_i` be the algebraic variety `\{x \in L^d \mid q_i(x) = 0\}`
        of `q_i` over the algebraic closure `L` of `K`.
        By [Raic2012]_, `f` can be written as

        .. MATH::

            (*) \quad \sum_A \frac{p_A}{\prod_{i \in A} q_i^{b_i}},

        where the `b_i` are positive integers, each `p_A` is a products
        of `p` and an element in `K[X]`,
        and the sum is taken over all subsets
        `A \subseteq \{1, \ldots, m\}` such that `|A| \leq d` and
        `\{q_i \mid i \in A\}` is algebraically independent.

        We call `(*)` an *algebraic dependence decomposition* of `f`.
        Algebraic dependence decompositions are not unique.

        The algorithm used comes from [Raic2012]_.

        OUTPUT:

        An instance of :class:`FractionWithFactoredDenominatorSum`.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: f = 1/(x^2 * (x*y + 1) * y^3)
            sage: ff = FFPD(f)
            sage: decomp = ff.algebraic_dependence_decomposition()
            sage: decomp
            (0, []) + (-x, [(x*y + 1, 1)]) +
            (x^2*y^2 - x*y + 1, [(y, 3), (x, 2)])
            sage: decomp.sum().quotient() == f
            True
            sage: for r in decomp:
            ....:     J = r.algebraic_dependence_certificate()
            ....:     J is None or J == J.ring().ideal()  # The zero ideal
            True
            True
            True

        ::

            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: G = sin(x)
            sage: H = x^2 * (x*y + 1) * y^3
            sage: f = FFPD(G, H.factor())
            sage: decomp = f.algebraic_dependence_decomposition()
            sage: decomp
            (0, []) + (x^4*y^3*sin(x), [(x*y + 1, 1)]) +
            (-(x^5*y^5 - x^4*y^4 + x^3*y^3 - x^2*y^2 + x*y - 1)*sin(x),
             [(y, 3), (x, 2)])
            sage: bool(decomp.sum().quotient() == G/H)
            True
            sage: for r in decomp:
            ....:     J = r.algebraic_dependence_certificate()
            ....:     J is None or J == J.ring().ideal()
            True
            True
            True
        """
        J = self.algebraic_dependence_certificate()
        if not J:
            # No decomposing possible.
            return FractionWithFactoredDenominatorSum([self])

        # Otherwise decompose recursively.
        decomp = FractionWithFactoredDenominatorSum()
        p = self.numerator()
        df = self.denominator_factored()
        m = len(df)
        g = J.gens()[0]     # An annihilating polynomial for df.
        new_vars = J.ring().gens()
        # Note that each new_vars[j] corresponds to df[j] such that
        # g([q**e for q, e in df]) = 0.
        # Assuming here that g.parent() has negdeglex term order
        # so that g.lt() is indeed the monomial we want below.
        # Use g to rewrite r into a sum of FFPDs,
        # each with < m distinct denominator factors.
        gg = (g.lt() - g) / (g.lc())
        numers = map(prod, zip(gg.coefficients(), gg.monomials()))
        e = list(g.lt().exponents())[0: m]
        denoms = [(new_vars[j], e[0][j] + 1) for j in range(m)]
        # Write r in terms of new_vars,
        # cancel factors in the denominator, and combine like terms.
        FFPD = FractionWithFactoredDenominatorRing(J.ring())
        iteration1_temp = FractionWithFactoredDenominatorSum(
            [FFPD(a, denoms) for a in numers])._combine_like_terms_()
        # Substitute in df.
        qpowsub = {new_vars[j]: df[j][0] ** df[j][1] for j in range(m)}
        iteration1 = FractionWithFactoredDenominatorSum()
        for r in iteration1_temp:
            num1 = p * J.ring()(r.numerator()).subs(qpowsub)
            denoms1 = []
            for q, e in r.denominator_factored():
                j = new_vars.index(q)
                denoms1.append((df[j][0], df[j][1] * e))
            iteration1.append(self.parent()(num1, denoms1))
        # Now decompose each FFPD of iteration1.
        for r in iteration1:
            decomp.extend(r.algebraic_dependence_decomposition())

        # Simplify and return result.
        return decomp._combine_like_terms_().whole_and_parts()

    def leinartas_decomposition(self):
        r"""
        Return a Leinartas decomposition of ``self``.

        Let `f = p/q` where `q` lies in a `d` -variate polynomial
        ring `K[X]` for some field `K`.
        Let `q_1^{e_1} \cdots q_n^{e_n}` be the
        unique factorization of `q` in `K[X]` into irreducible factors and
        let `V_i` be the algebraic variety
        `\{x\in L^d \mid q_i(x) = 0\}` of `q_i` over the algebraic closure
        `L` of `K`. By [Raic2012]_, `f` can be written as

        .. MATH::

            (*) \quad \sum_A \frac{p_A}{\prod_{i \in A} q_i^{b_i}},

        where the `b_i` are positive integers, each `p_A` is a product of
        `p` and an element of `K[X]`, and the sum is taken over all
        subsets `A \subseteq \{1, \ldots, m\}` such that

        1. `|A| \le d`,
        2. `\bigcap_{i\in A} T_i \neq \emptyset`, and
        3. `\{q_i \mid i\in A\}` is algebraically independent.

        In particular, any rational expression in `d` variables
        can be represented as a sum of rational expressions
        whose denominators each contain at most `d` distinct irreducible
        factors.

        We call `(*)` a *Leinartas decomposition* of `f`.
        Leinartas decompositions are not unique.

        The algorithm used comes from [Raic2012]_.

        OUTPUT:

        An instance of :class:`FractionWithFactoredDenominatorSum`.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: f = (x^2 + 1)/((x + 2)*(x - 1)*(x^2 + x + 1))
            sage: decomp = FFPD(f).leinartas_decomposition()
            sage: decomp
            (0, []) + (2/9, [(x - 1, 1)]) +
            (-5/9, [(x + 2, 1)]) + (1/3*x, [(x^2 + x + 1, 1)])
            sage: decomp.sum().quotient() == f
            True

        ::

            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: f = 1/x + 1/y + 1/(x*y + 1)
            sage: decomp = FFPD(f).leinartas_decomposition()
            sage: decomp
            (0, []) + (1, [(x*y + 1, 1)]) + (x + y, [(y, 1), (x, 1)])
            sage: decomp.sum().quotient() == f
            True
            sage: def check_decomp(r):
            ....:     L = r.nullstellensatz_certificate()
            ....:     J = r.algebraic_dependence_certificate()
            ....:     return L is None and (J is None or J == J.ring().ideal())
            sage: all(check_decomp(r) for r in decomp)
            True

        ::

            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: f = sin(x)/x + 1/y + 1/(x*y + 1)
            sage: G = f.numerator()
            sage: H = R(f.denominator())
            sage: ff = FFPD(G, H.factor())
            sage: decomp = ff.leinartas_decomposition()
            sage: decomp
            (0, []) +
            (-(x*y^2*sin(x) + x^2*y + x*y + y*sin(x) + x)*y, [(y, 1)]) +
            ((x*y^2*sin(x) + x^2*y + x*y + y*sin(x) + x)*x*y, [(x*y + 1, 1)]) +
            (x*y^2*sin(x) + x^2*y + x*y + y*sin(x) + x, [(y, 1), (x, 1)])
            sage: bool(decomp.sum().quotient() == f)
            True
            sage: all(check_decomp(r) for r in decomp)
            True

        ::

            sage: R.<x,y,z>= PolynomialRing(GF(2, 'a'))
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: f = 1/(x * y * z * (x*y + z))
            sage: decomp = FFPD(f).leinartas_decomposition()
            sage: decomp
            (0, []) + (1, [(z, 2), (x*y + z, 1)]) +
            (1, [(z, 2), (y, 1), (x, 1)])
            sage: decomp.sum().quotient() == f
            True
        """
        if self.dimension() == 1:
            # Sage's lift() function doesn't work in
            # univariate polynomial rings.
            # So nullstellensatz_decomposition() won't work.
            # Can use algebraic_dependence_decomposition(),
            # which is sufficient.
            # temp = FractionWithFactoredDenominatorSum([self])
            # Alternatively can use univariate_decomposition(),
            # which is more efficient.
            return self.univariate_decomposition()
        temp = self.nullstellensatz_decomposition()
        decomp = FractionWithFactoredDenominatorSum()
        for r in temp:
            decomp.extend(r.algebraic_dependence_decomposition())

        # Simplify and return result.
        return decomp._combine_like_terms_().whole_and_parts()

    def cohomology_decomposition(self):
        r"""
        Return the cohomology decomposition of ``self``.

        Let `p / (q_1^{e_1} \cdots q_n^{e_n})` be the fraction represented
        by ``self`` and let `K[x_1, \ldots, x_d]` be the polynomial ring
        in which the `q_i` lie.
        Assume that `n \leq d` and that the gradients of the `q_i` are linearly
        independent at all points in the intersection
        `V_1 \cap \ldots \cap V_n` of the algebraic varieties
        `V_i = \{x \in L^d \mid q_i(x) = 0 \}`, where `L` is the algebraic
        closure of the field `K`.
        Return a :class:`FractionWithFactoredDenominatorSum`
        `f` such that the differential form
        `f dx_1 \wedge \cdots \wedge dx_d` is de Rham cohomologous to the
        differential form
        `p / (q_1^{e_1} \cdots q_n^{e_n}) dx_1 \wedge \cdots \wedge dx_d`
        and such that the denominator of each summand of `f` contains
        no repeated irreducible factors.

        The algorithm used here comes from the proof of Theorem 17.4 of
        [AiYu1983]_.

        OUTPUT:

        An instance of :class:`FractionWithFactoredDenominatorSum`.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: f = 1/(x^2 + x + 1)^3
            sage: decomp = FFPD(f).cohomology_decomposition()
            sage: decomp
            (0, []) + (2/3, [(x^2 + x + 1, 1)])

            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: FFPD(1, [(x, 1), (y, 2)]).cohomology_decomposition()
            (0, [])

            sage: p = 1
            sage: qs = [(x*y - 1, 1), (x**2 + y**2 - 1, 2)]
            sage: f = FFPD(p, qs)
            sage: f.cohomology_decomposition()
            (0, []) + (4/3*x*y + 4/3, [(x^2 + y^2 - 1, 1)]) +
            (1/3, [(x*y - 1, 1), (x^2 + y^2 - 1, 1)])
        """
        from sage.calculus.functions import jacobian
        from sage.arith.all import xgcd
        from sage.sets.set import Set

        R = self.denominator_ring
        df = self.denominator_factored()
        n = len(df)
        if sum(e for (q, e) in df) <= n:
            # No decomposing possible.
            return FractionWithFactoredDenominatorSum([self])

        # Otherwise decompose recursively.
        decomp = FractionWithFactoredDenominatorSum()
        p = self.numerator()
        qs = [q for (q, e) in df]
        X = sorted(R.gens())
        var_sets_n = Set(X).subsets(n)
        Par = self.parent()

        # Compute Jacobian determinants for qs.
        dets = []
        for v in var_sets_n:
            # Sort v according to the term order of R.
            x = sorted(v)
            jac = jacobian(qs, x)
            dets.append(R(jac.determinant()))

        # Get a Nullstellensatz certificate for qs and dets.
        if self.dimension() == 1:
            # Sage's lift() function doesn't work in
            # univariate polynomial rings.
            # So use xgcd(), which does the same thing in this case.
            # Note that by assumption qs and dets have length 1.
            L = xgcd(qs[0], dets[0])[1:]
        else:
            L = R.one().lift(R.ideal(qs + dets))

        # Do first iteration of decomposition.
        iteration1 = FractionWithFactoredDenominatorSum()
        # Contributions from qs.
        for i in range(n):
            if L[i] == 0:
                continue
            # Cancel one df[i] from denominator.
            new_df = [list(t) for t in df]
            if new_df[i][1] > 1:
                new_df[i][1] -= 1
            else:
                del new_df[i]
            iteration1.append(Par(p * L[i], new_df))

        # Contributions from dets.
        # Compute each contribution's cohomologous form using
        # the least index j such that new_df[j][1] > 1.
        # Know such an index exists by first 'if' statement at
        # the top.
        for j in range(n):
            if df[j][1] > 1:
                J = j
                break
        new_df = [list(t) for t in df]
        new_df[J][1] -= 1
        for k in range(var_sets_n.cardinality()):
            if L[n + k] == 0:
                continue
            # Sort variables according to the term order of R.
            x = sorted(var_sets_n[k])
            # Compute Jacobian in the Symbolic Ring.
            jac = jacobian([SR(p * L[n + k])] +
                           [SR(qs[j]) for j in range(n) if j != J],
                           [SR(xx) for xx in x])
            det = jac.determinant()
            psign = permutation_sign(x, X)
            iteration1.append(Par((-1) ** J * det / (psign * new_df[J][1]),
                                  new_df))

        # Now decompose each FFPD of iteration1.
        for r in iteration1:
            decomp.extend(r.cohomology_decomposition())

        # Simplify and return result.
        return decomp._combine_like_terms_().whole_and_parts()

    def asymptotic_decomposition(self, alpha, asy_var=None):
        r"""
        Return the asymptotic decomposition of ``self``.

        The asymptotic decomposition of `F` is a sum that has the
        same asymptotic expansion as `f` in the direction ``alpha``
        but each summand has a denominator factorization of the form
        `[(q_1, 1), \ldots, (q_n, 1)]`, where `n` is at most the
        :meth:`dimension` of `F`.

        INPUT:

        - ``alpha`` -- a `d`-tuple of positive integers or symbolic variables
        - ``asy_var`` -- (default: ``None``) a symbolic variable with
          respect to which to compute asymptotics;
          if ``None`` is given, we set ``asy_var = var('r')``

        OUTPUT:

        An instance of :class:`FractionWithFactoredDenominatorSum`.

        The output results from a Leinartas decomposition followed by a
        cohomology decomposition.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: f = (x^2 + 1)/((x - 1)^3*(x + 2))
            sage: F = FFPD(f)
            sage: alpha = [var('a')]
            sage: F.asymptotic_decomposition(alpha)
            (0, []) +
            (1/54*(5*a^2*x^2 + 2*a^2*x + 11*a^2)*r^2/x^2
             - 1/54*(5*a*x^2 - 2*a*x - 33*a)*r/x^2 + 11/27/x^2, [(x - 1, 1)]) +
            (-5/27, [(x + 2, 1)])

        ::

            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: H = (1 - 2*x -y)*(1 - x -2*y)**2
            sage: Hfac = H.factor()
            sage: G = 1/Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: alpha = var('a, b')
            sage: F.asymptotic_decomposition(alpha)
            (0, []) +
            (1/3*(2*b*x - a*y)*r/(x*y) + 1/3*(2*x - y)/(x*y),
             [(x + 2*y - 1, 1), (2*x + y - 1, 1)])
        """
        R = self.denominator_ring
        d = self.dimension()
        n = len(self.denominator_factored())
        X = [SR(x) for x in R.gens()]

        # Reduce number of distinct factors in denominator of self
        # down to at most d.
        decomp1 = FractionWithFactoredDenominatorSum([self])
        if n > d:
            decomp1 = decomp1[0].leinartas_decomposition()

        # Reduce to no repeated factors in denominator of each element
        # of decomp1.
        # Compute the cohomology decomposition for each
        # Cauchy differential form generated by each element of decomp.
        if asy_var is None:
            asy_var = var('r')
        cauchy_stuff = prod([X[j] ** (-alpha[j] * asy_var - 1)
                             for j in range(d)])
        decomp2 = FractionWithFactoredDenominatorSum()
        for f in decomp1:
            ff = self.parent()(f.numerator() * cauchy_stuff,
                               f.denominator_factored())
            decomp2.extend(ff.cohomology_decomposition())
        decomp2 = decomp2._combine_like_terms_()

        # Divide out cauchy_stuff from integrands.
        decomp3 = FractionWithFactoredDenominatorSum()
        for f in decomp2:
            ff = self.parent()((f.numerator() /
                                cauchy_stuff).simplify_full().collect(asy_var),
                      f.denominator_factored())
            decomp3.append(ff)

        return decomp3

    def asymptotics(self, p, alpha, N, asy_var=None, numerical=0,
                    verbose=False):
        r"""
        Return the asymptotics in the given direction.

        This function returns the first `N` terms (some of which could be
        zero) of the asymptotic expansion of the Maclaurin ray coefficients
        `F_{r \alpha}` of the function `F` represented by ``self`` as
        `r \to \infty`, where `r` is ``asy_var`` and ``alpha`` is a tuple
        of positive integers of length `d` which is ``self.dimension()``.
        Assume that

        - `F` is holomorphic in a neighborhood of the origin;
        - the unique factorization of the denominator `H` of `F` in the local
          algebraic ring at `p` equals its unique factorization in the local
          analytic ring at `p`;
        - the unique factorization of `H` in the local algebraic ring at `p`
          has at most ``d`` irreducible factors, none of which are repeated
          (one can reduce to this case via :meth:`asymptotic_decomposition()`);
        - `p` is a convenient strictly minimal smooth or multiple point
          with all nonzero coordinates that is critical and nondegenerate
          for ``alpha``.

        The algorithms used here come from [RaWi2008a]_ and [RaWi2012]_.

        INPUT:

        - ``p`` -- a dictionary with keys that can be coerced to equal
          ``self.denominator_ring.gens()``
        - ``alpha`` -- a tuple of length ``self.dimension()`` of
          positive integers or, if `p` is a smooth point,
          possibly of symbolic variables
        - ``N`` -- a positive integer
        - ``asy_var`` -- (default: ``None``) a symbolic variable for the
          asymptotic expansion; if ``none`` is given, then
          ``var('r')`` will be assigned
        - ``numerical`` -- (default: 0) a natural number;
          if ``numerical`` is greater than 0, then return a numerical
          approximation of `F_{r \alpha}` with ``numerical`` digits of
          precision; otherwise return exact values
        - ``verbose`` -- (default: ``False``) print the current state of
          the algorithm

        OUTPUT:

        The tuple ``(asy, exp_scale, subexp_part)``.
        Here ``asy`` is the sum of the first `N` terms (some of which might
        be 0) of the asymptotic expansion of `F_{r\alpha}` as `r \to \infty`;
        ``exp_scale**r`` is the exponential factor of ``asy``;
        ``subexp_part`` is the subexponential factor of ``asy``.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing

        A smooth point example::

            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: H = (1 - x - y - x*y)**2
            sage: Hfac = H.factor()
            sage: G = 1/Hfac.unit()
            sage: F = FFPD(G, Hfac); print(F)
            (1, [(x*y + x + y - 1, 2)])
            sage: alpha = [4, 3]
            sage: decomp = F.asymptotic_decomposition(alpha); decomp
            (0, []) + (-3/2*r*(y + 1)/y - 1/2*(y + 1)/y, [(x*y + x + y - 1, 1)])
            sage: F1 = decomp[1]
            sage: p = {y: 1/3, x: 1/2}
            sage: asy = F1.asymptotics(p, alpha, 2, verbose=True)
            Creating auxiliary functions...
            Computing derivatives of auxiliary functions...
            Computing derivatives of more auxiliary functions...
            Computing second order differential operator actions...
            sage: asy
            (1/6000*(3600*sqrt(5)*sqrt(3)*sqrt(2)*sqrt(r)/sqrt(pi)
              + 463*sqrt(5)*sqrt(3)*sqrt(2)/(sqrt(pi)*sqrt(r)))*432^r,
             432,
             3/5*sqrt(5)*sqrt(3)*sqrt(2)*sqrt(r)/sqrt(pi)
              + 463/6000*sqrt(5)*sqrt(3)*sqrt(2)/(sqrt(pi)*sqrt(r)))
            sage: F.relative_error(asy[0], alpha, [1, 2, 4, 8, 16], asy[1])
            [((4, 3), 2.083333333, [2.092576110], [-0.0044365330...]),
             ((8, 6), 2.787374614, [2.790732875], [-0.0012048112...]),
             ((16, 12), 3.826259447, [3.827462310], [-0.0003143703...]),
             ((32, 24), 5.328112821, [5.328540787], [-0.0000803222...]),
             ((64, 48), 7.475927885, [7.476079664], [-0.0000203023...])]

        A multiple point example::

            sage: R.<x,y,z>= PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: H = (4 - 2*x - y - z)**2*(4 - x - 2*y - z)
            sage: Hfac = H.factor()
            sage: G = 16/Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: F
            (-16, [(x + 2*y + z - 4, 1), (2*x + y + z - 4, 2)])
            sage: alpha = [3, 3, 2]
            sage: decomp = F.asymptotic_decomposition(alpha); decomp
            (0, []) +
            (16*r*(4*y - 3*z)/(y*z) + 16*(2*y - z)/(y*z),
             [(x + 2*y + z - 4, 1), (2*x + y + z - 4, 1)])
            sage: F1 = decomp[1]
            sage: p = {x: 1, y: 1, z: 1}
            sage: asy = F1.asymptotics(p, alpha, 2, verbose=True) # long time
            Creating auxiliary functions...
            Computing derivatives of auxiliary functions...
            Computing derivatives of more auxiliary functions...
            Computing second-order differential operator actions...
            sage: asy # long time
            (4/3*sqrt(3)*sqrt(r)/sqrt(pi) + 47/216*sqrt(3)/(sqrt(pi)*sqrt(r)),
             1, 4/3*sqrt(3)*sqrt(r)/sqrt(pi) + 47/216*sqrt(3)/(sqrt(pi)*sqrt(r)))
            sage: F.relative_error(asy[0], alpha, [1, 2, 4, 8], asy[1]) # long time
            [((3, 3, 2), 0.9812164307, [1.515572606], [-0.54458543...]),
             ((6, 6, 4), 1.576181132, [1.992989399], [-0.26444185...]),
             ((12, 12, 8), 2.485286378, [2.712196351], [-0.091301338...]),
             ((24, 24, 16), 3.700576827, [3.760447895], [-0.016178847...])]
        """
        R = self.denominator_ring

        # Coerce keys of p into R.
        p = coerce_point(R, p)

        if asy_var is None:
            asy_var = var('r')
        d = self.dimension()
        X = list(R.gens())
        alpha = list(alpha)
        df = self.denominator_factored()
        n = len(df)     # Number of smooth factors

        # Find greatest i such that X[i] is a convenient coordinate,
        # that is, such that for all (h, e) in df, we have
        # (X[i]*diff(h, X[i])).subs(p) != 0.
        # Assuming such an i exists.
        i = d - 1
        while 0 in [(X[i] * diff(h, X[i])).subs(p) for (h, e) in df]:
            i -= 1
        coordinate = i

        if n == 1:
            # Smooth point.
            return self.asymptotics_smooth(p, alpha, N, asy_var, coordinate,
                                           numerical, verbose=verbose)

        # Multiple point.
        return self.asymptotics_multiple(p, alpha, N, asy_var, coordinate,
                                         numerical, verbose=verbose)

    def asymptotics_smooth(self, p, alpha, N, asy_var, coordinate=None,
                           numerical=0, verbose=False):
        r"""
        Return the asymptotics in the given direction of a smooth point.

        This is the same as :meth:`asymptotics()`, but only in the
        case of a convenient smooth point.

        The formulas used for computing the asymptotic expansions are
        Theorems 3.2 and 3.3 [RaWi2008a]_ with the exponent of `H`
        equal to 1. Theorem 3.2 is a specialization of Theorem 3.4
        of [RaWi2012]_ with `n = 1`.

        INPUT:

        - ``p`` -- a dictionary with keys that can be coerced to equal
          ``self.denominator_ring.gens()``
        - ``alpha`` -- a tuple of length ``d = self.dimension()`` of
          positive integers or, if `p` is a smooth point,
          possibly of symbolic variables
        - ``N`` -- a positive integer
        - ``asy_var`` -- (optional; default: ``None``) a symbolic variable;
          the variable of the asymptotic expansion,
          if none is given, ``var('r')`` will be assigned
        - ``coordinate`` -- (optional; default: ``None``) an integer in
          `\{0, \ldots, d-1\}` indicating a convenient coordinate to base
          the asymptotic calculations on; if ``None`` is assigned, then
          choose ``coordinate=d-1``
        - ``numerical`` -- (optional; default: 0) a natural number;
          if numerical is greater than 0, then return a numerical approximation
          of the Maclaurin ray coefficients of ``self`` with ``numerical``
          digits of precision; otherwise return exact values
        - ``verbose`` -- (default: ``False``) print the current state of
          the algorithm

        OUTPUT:

        The asymptotic expansion.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: H = 2 - 3*x
            sage: Hfac = H.factor()
            sage: G = 1/Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: F
            (-1/3, [(x - 2/3, 1)])
            sage: alpha = [2]
            sage: p = {x: 2/3}
            sage: asy = F.asymptotics_smooth(p, alpha, 3, asy_var=var('r'))
            sage: asy
            (1/2*(9/4)^r, 9/4, 1/2)

        ::

            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: H = 1-x-y-x*y
            sage: Hfac = H.factor()
            sage: G = 1/Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: alpha = [3, 2]
            sage: p = {y: 1/2*sqrt(13) - 3/2, x: 1/3*sqrt(13) - 2/3}
            sage: F.asymptotics_smooth(p, alpha, 2, var('r'), numerical=3, verbose=True)
            Creating auxiliary functions...
            Computing derivatives of auxiliary functions...
            Computing derivatives of more auxiliary functions...
            Computing second order differential operator actions...
            (71.2^r*(0.369/sqrt(r) - 0.018.../r^(3/2)), 71.2, 0.369/sqrt(r) - 0.018.../r^(3/2))

            sage: q = 1/2
            sage: qq = q.denominator()
            sage: H = 1 - q*x + q*x*y - x^2*y
            sage: Hfac = H.factor()
            sage: G = (1 - q*x)/Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: alpha = list(qq*vector([2, 1 - q]))
            sage: alpha
            [4, 1]
            sage: p = {x: 1, y: 1}
            sage: F.asymptotics_smooth(p, alpha, 5, var('r'), verbose=True) # not tested (140 seconds)
            Creating auxiliary functions...
            Computing derivatives of auxiliary functions...
            Computing derivatives of more auxiliary functions...
            Computing second order differential operator actions...
            (1/12*sqrt(3)*2^(2/3)*gamma(1/3)/(pi*r^(1/3))
              - 1/96*sqrt(3)*2^(1/3)*gamma(2/3)/(pi*r^(5/3)),
             1,
             1/12*sqrt(3)*2^(2/3)*gamma(1/3)/(pi*r^(1/3))
              - 1/96*sqrt(3)*2^(1/3)*gamma(2/3)/(pi*r^(5/3)))
        """
        from sage.calculus.functions import jacobian
        from sage.calculus.var import function
        from sage.functions.other import factorial, gamma, sqrt
        from sage.functions.log import exp, log
        from sage.matrix.constructor import matrix
        from sage.modules.free_module_element import vector
        from sage.symbolic.constants import pi
        from sage.symbolic.relation import solve
        from sage.rings.all import CC
        from sage.rings.rational_field import QQ

        R = self.denominator_ring
        d = self.dimension()
        I = sqrt(-ZZ.one())
        # Coerce everything into the Symbolic Ring.
        X = [SR(x) for x in R.gens()]
        G = SR(self.numerator())
        H = SR(self.denominator())
        p = {SR(x): p[x] for x in R.gens()}
        alpha = [SR(a) for a in alpha]

        # Put given convenient coordinate at end of variable list.
        if coordinate is not None:
            x = X.pop(coordinate)
            X.append(x)
            a = alpha.pop(coordinate)
            alpha.append(a)

        # Deal with the simple univariate case first.
        # Same as the multiple point case with n == d.
        # but with a negative sign.
        # I'll just past the code from the multiple point case.
        if d == 1:
            det = jacobian(H, X).subs(p).determinant().abs()
            exp_scale = prod([(p[X[i]] ** (-alpha[i])).subs(p)
                              for i in range(d)])
            subexp_part = -G.subs(p) / (det * prod(p.values()))
            if numerical:
                exp_scale = exp_scale.n(digits=numerical)
                subexp_part = subexp_part.n(digits=numerical)
            return (exp_scale ** asy_var * subexp_part, exp_scale, subexp_part)

        # If p is a tuple of rationals, then compute with it directly.
        # Otherwise, compute symbolically and plug in p at the end.
        if vector(p.values()) in QQ ** d:
            P = p
        else:
            sP = [var('p' + str(j)) for j in range(d)]
            P = {X[j]: sP[j] for j in range(d)}
            p = {sP[j]: p[X[j]] for j in range(d)}

        # Setup.
        if verbose:
            print("Creating auxiliary functions...")
        # Implicit functions.
        h = function('h')(*tuple(X[:d - 1]))
        U = function('U')(*tuple(X))
        # All other functions are defined in terms of h, U, and
        # explicit functions.
        Gcheck = -G / U * (h / X[d - 1])
        A = Gcheck.subs({X[d - 1]: ZZ.one() / h}) / h
        t = 't'
        L = [str(elt) for elt in X]
        while t in L:
            t = t + 't'
        T = [var(t + str(i)) for i in range(d - 1)]
        e = {X[i]: P[X[i]] * exp(I * T[i]) for i in range(d - 1)}
        ht = h.subs(e)
        At = A.subs(e)
        Phit = (-log(P[X[d - 1]] * ht) +
                I * sum([alpha[i] / alpha[d - 1] * T[i]
                         for i in range(d - 1)]))
        Tstar = {t: ZZ.zero() for t in T}
        # Store h and U and all their derivatives evaluated at P.
        atP = P.copy()
        atP.update({h.subs(P): ZZ.one() / P[X[d - 1]]})

        # Compute the derivatives of h up to order 2 * N, evaluate at P,
        # and store in atP.
        # Keep a copy of unevaluated h derivatives for use in the case
        # d = 2 and v > 2 below.
        hderivs1 = {}   # First derivatives of h.
        for i in range(d - 1):
            s = solve(diff(H.subs({X[d - 1]: ZZ.one() / h}), X[i]),
                      diff(h, X[i]))[0].rhs().simplify()
            hderivs1.update({diff(h, X[i]): s})
            atP.update({diff(h, X[i]).subs(P): s.subs(P).subs(atP)})
        hderivs = diff_all(h, X[0: d - 1], 2 * N, sub=hderivs1, rekey=h)
        for k in hderivs:
            atP.update({k.subs(P): hderivs[k].subs(atP)})

        # Compute the derivatives of U up to order 2 * N and evaluate at P.
        # To do this, differentiate H = U*Hcheck over and over, evaluate at P,
        # and solve for the derivatives of U at P.
        # Need the derivatives of H with short keys to pass on
        # to diff_prod later.
        Hderivs = diff_all(H, X, 2 * N, ending=[X[d - 1]], sub_final=P)
        if verbose:
            print("Computing derivatives of auxiliary functions...")
        # For convenience in checking if all the nontrivial derivatives of U
        # at p are zero a few line below, store the value of U(p) in atP
        # instead of in Uderivs.
        Uderivs = {}
        atP.update({U.subs(P): diff(H, X[d - 1]).subs(P)})
        end = [X[d - 1]]
        Hcheck = X[d - 1] - ZZ.one() / h
        k = H.polynomial(CC).degree() - 1
        if k == 0:
            # Then we can conclude that all higher derivatives of U are zero.
            for l in range(1, 2 * N + 1):
                for s in combinations_with_replacement(X, l):
                    Uderivs[diff(U, list(s)).subs(P)] = ZZ.zero()
        elif k > 0 and k < 2 * N:
            all_zero = True
            Uderivs = diff_prod(Hderivs, U, Hcheck, X,
                                range(1, k + 1), end, Uderivs, atP)
            # Check for a nonzero U derivative.
            if any(u for u in Uderivs.values()):
                all_zero = False
            if all_zero:
                # Then, using a proposition at the end of [RaWi2012], we can
                # conclude that all higher derivatives of U are zero.
                for l in range(k + 1, 2 * N + 1):
                    for s in combinations_with_replacement(X, l):
                        Uderivs.update({diff(U, list(s)).subs(P): ZZ.zero()})
            else:
                # Have to compute the rest of the derivatives.
                Uderivs = diff_prod(Hderivs, U, Hcheck, X,
                                    range(k + 1, 2 * N + 1), end, Uderivs, atP)
        else:
            Uderivs = diff_prod(Hderivs, U, Hcheck, X,
                                range(1, 2 * N + 1), end, Uderivs, atP)
        atP.update(Uderivs)

        # In general, this algorithm is not designed to handle the case of a
        # singular Phit''(Tstar).
        # However, when d = 2 the algorithm can cope.
        if d == 2:
            # Compute v, the order of vanishing at Tstar of Phit.
            # It is at least 2.
            v = Integer(2)
            Phitderiv = diff(Phit, T[0], 2)
            splat = Phitderiv.subs(Tstar).subs(atP).subs(p).simplify()
            while splat == 0:
                v += 1
                if v > 2 * N:
                    # Then need to compute more derivatives of h for atP.
                    hderivs.update({diff(h, X[0], v):
                                    diff(hderivs[diff(h, X[0], v - 1)],
                                    X[0]).subs(hderivs1)})
                    atP.update({diff(h, X[0], v).subs(P):
                                hderivs[diff(h, X[0], v)].subs(atP)})
                Phitderiv = diff(Phitderiv, T[0])
                splat = Phitderiv.subs(Tstar).subs(atP).subs(p).simplify()

        if d == 2 and v > 2:
            t = T[0]  # Simplify variable names.
            a = splat / factorial(v)
            Phitu = Phit - a * t ** v

            # Compute all partial derivatives of At and Phitu
            # up to orders 2*(N - 1) and 2*(N - 1) + v, respectively,
            # in case v is even.
            # Otherwise, compute up to orders N - 1 and N - 1 + v,
            # respectively.
            # To speed up later computations,
            # create symbolic functions AA and BB
            # to stand in for the expressions At and Phitu, respectively.
            if verbose:
                print("Computing derivatives of more auxiliary functions...")
            AA = function('AA')(t)
            BB = function('BB')(t)
            if v.mod(2) == 0:
                At_derivs = diff_all(At, T, 2 * N - 2, sub=hderivs1,
                                     sub_final=[Tstar, atP], rekey=AA)
                Phitu_derivs = diff_all(Phitu, T, 2 * N - 2 +v,
                                        sub=hderivs1, sub_final=[Tstar, atP],
                                        zero_order=v + 1, rekey=BB)
            else:
                At_derivs = diff_all(At, T, N - 1, sub=hderivs1,
                                     sub_final=[Tstar, atP], rekey=AA)
                Phitu_derivs = diff_all(Phitu, T, N - 1 + v,
                                        sub=hderivs1, sub_final=[Tstar, atP],
                                        zero_order=v + 1 , rekey=BB)
            AABB_derivs = At_derivs
            AABB_derivs.update(Phitu_derivs)
            AABB_derivs[AA] = At.subs(Tstar).subs(atP)
            AABB_derivs[BB] = Phitu.subs(Tstar).subs(atP)
            if verbose:
                print("Computing second order differential operator actions...")
            DD = diff_op_simple(AA, BB, AABB_derivs, t, v, a, N)

            # Plug above into asymptotic formula.
            L = []
            if v.mod(2) == 0:
                for k in range(N):
                    L.append(sum([(-1) ** l * gamma((2 * k + v * l + 1) / v) /
                                  (factorial(l) * factorial(2 * k + v * l)) *
                                  DD[(k, l)] for l in range(0, 2 * k + 1)]))
                chunk = (a ** (-1 / v) / (pi * v) *
                         sum([alpha[d - 1] ** (-(2 * k + 1) / v) *
                              L[k] * asy_var ** (-(2 * k + 1) / v)
                              for k in range(N)]))
            else:
                zeta = exp(I * pi / (2 * v))
                for k in range(N):
                    L.append(sum([(-1) ** l * gamma((k + v * l + 1) / v) /
                                  (factorial(l) * factorial(k + v * l)) *
                                  (zeta ** (k + v * l + 1) +
                                   (-1) ** (k + v * l) *
                                   zeta ** (-(k + v * l + 1))) *
                                  DD[(k, l)] for l in range(0, k + 1)]))
                chunk = (abs(a) ** (-1 / v) / (2 * pi * v) *
                         sum([alpha[d - 1] ** (-(k + 1) / v) *
                              L[k] * asy_var ** (-(k + 1) / v)
                              for k in range(N)]))

        # Asymptotics for d >= 2 case.
        # A singular Phit''(Tstar) will cause a crash in this case.
        else:
            Phit1 = jacobian(Phit, T).subs(hderivs1)
            a = jacobian(Phit1, T).subs(hderivs1).subs(Tstar).subs(atP)
            a_inv = a.inverse()
            Phitu = (Phit - (1 / QQ(2)) * matrix([T]) *
                     a * matrix([T]).transpose())
            Phitu = Phitu[0][0]
            # Compute all partial derivatives of At and Phitu up to
            # orders 2 * N-2 and 2 * N, respectively.
            # Take advantage of the fact that At and Phitu
            # are sufficiently differentiable functions so that mixed partials
            # are equal.  Thus only need to compute representative partials.
            # Choose nondecreasing sequences as representative differentiation-
            # order sequences.
            # To speed up later computations,
            # create symbolic functions AA and BB
            # to stand in for the expressions At and Phitu, respectively.
            if verbose:
                print("Computing derivatives of more auxiliary functions...")
            AA = function('AA')(*tuple(T))
            At_derivs = diff_all(At, T, 2 * N - 2, sub=hderivs1,
                                 sub_final=[Tstar, atP], rekey=AA)
            BB = function('BB')(*tuple(T))
            Phitu_derivs = diff_all(Phitu, T, 2 * N, sub=hderivs1,
                                    sub_final=[Tstar, atP], rekey=BB, zero_order=3)
            AABB_derivs = At_derivs
            AABB_derivs.update(Phitu_derivs)
            AABB_derivs[AA] = At.subs(Tstar).subs(atP)
            AABB_derivs[BB] = Phitu.subs(Tstar).subs(atP)
            if verbose:
                print("Computing second order differential operator actions...")
            DD = diff_op(AA, BB, AABB_derivs, T, a_inv, 1 , N)

            # Plug above into asymptotic formula.
            L = []
            for k in range(N):
                L.append(sum([DD[(0, k, l)] / ((-1) ** k * 2 ** (l + k) *
                                               factorial(l) * factorial(l + k))
                              for l in range(0, 2 * k + 1)]))
            chunk = sum([(2 * pi) ** ((1 - d) / Integer(2)) *
                         a.determinant() ** (-ZZ.one() / Integer(2)) *
                         alpha[d - 1] ** ((ZZ.one() - d) / Integer(2) - k) *
                         L[k] *
                         asy_var ** ((ZZ.one() - d) / Integer(2) - k)
                         for k in range(N)])

        chunk = chunk.subs(p).simplify()
        coeffs = chunk.coefficients(asy_var)
        coeffs.reverse()
        coeffs = coeffs[:N]
        if numerical:
            subexp_part = sum([co[0].subs(p).n(digits=numerical) *
                               asy_var ** co[1] for co in coeffs])
            exp_scale = prod([(P[X[i]] ** (-alpha[i])).subs(p)
                              for i in range(d)]).n(digits=numerical)
        else:
            subexp_part = sum([co[0].subs(p) * asy_var ** co[1]
                               for co in coeffs])
            exp_scale = prod([(P[X[i]] ** (-alpha[i])).subs(p)
                              for i in range(d)])
        return (exp_scale ** asy_var * subexp_part, exp_scale, subexp_part)

    def asymptotics_multiple(self, p, alpha, N, asy_var, coordinate=None,
                             numerical=0, verbose=False):
        r"""
        Return the asymptotics in the given direction of a multiple
        point nondegenerate for ``alpha``.

        This is the same as :meth:`asymptotics`, but only in the case
        of a convenient multiple point nondegenerate for ``alpha``.
        Assume also that ``self.dimension >= 2`` and that the
        ``p.values()`` are not symbolic variables.

        The formulas used for computing the asymptotic expansion are
        Theorem 3.4 and Theorem 3.7 of [RaWi2012]_.

        INPUT:

        - ``p`` -- a dictionary with keys that can be coerced to equal
          ``self.denominator_ring.gens()``
        - ``alpha`` -- a tuple of length ``d = self.dimension()`` of
          positive integers or, if `p` is a smooth point,
          possibly of symbolic variables
        - ``N`` -- a positive integer
        - ``asy_var`` -- (optional; default: ``None``) a symbolic variable;
          the variable of the asymptotic expansion,
          if none is given, ``var('r')`` will be assigned
        - ``coordinate`` -- (optional; default: ``None``) an integer in
          `\{0, \ldots, d-1\}` indicating a convenient coordinate to base
          the asymptotic calculations on; if ``None`` is assigned, then
          choose ``coordinate=d-1``
        - ``numerical`` -- (optional; default: 0) a natural number;
          if numerical is greater than 0, then return a numerical approximation
          of the Maclaurin ray coefficients of ``self`` with ``numerical``
          digits of precision; otherwise return exact values
        - ``verbose`` -- (default: ``False``) print the current state of
          the algorithm

        OUTPUT:

        The asymptotic expansion.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y,z>= PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: H = (4 - 2*x - y - z)*(4 - x -2*y - z)
            sage: Hfac = H.factor()
            sage: G = 16/Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: F
            (16, [(x + 2*y + z - 4, 1), (2*x + y + z - 4, 1)])
            sage: p = {x: 1, y: 1, z: 1}
            sage: alpha = [3, 3, 2]
            sage: F.asymptotics_multiple(p, alpha, 2, var('r'), verbose=True) # long time
            Creating auxiliary functions...
            Computing derivatives of auxiliary functions...
            Computing derivatives of more auxiliary functions...
            Computing second-order differential operator actions...
            (4/3*sqrt(3)/(sqrt(pi)*sqrt(r)) - 25/216*sqrt(3)/(sqrt(pi)*r^(3/2)),
             1,
             4/3*sqrt(3)/(sqrt(pi)*sqrt(r)) - 25/216*sqrt(3)/(sqrt(pi)*r^(3/2)))

            sage: H = (1 - x*(1 + y))*(1 - z*x**2*(1 + 2*y))
            sage: Hfac = H.factor()
            sage: G = 1/Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: F
            (1, [(x*y + x - 1, 1), (2*x^2*y*z + x^2*z - 1, 1)])
            sage: p = {x: 1/2, z: 4/3, y: 1}
            sage: alpha = [8, 3, 3]
            sage: F.asymptotics_multiple(p, alpha, 2, var('r'), coordinate=1, verbose=True) # long time
            Creating auxiliary functions...
            Computing derivatives of auxiliary functions...
            Computing derivatives of more auxiliary functions...
            Computing second-order differential operator actions...
            (1/172872*108^r*(24696*sqrt(7)*sqrt(3)/(sqrt(pi)*sqrt(r))
              - 1231*sqrt(7)*sqrt(3)/(sqrt(pi)*r^(3/2))),
             108,
             1/7*sqrt(7)*sqrt(3)/(sqrt(pi)*sqrt(r))
              - 1231/172872*sqrt(7)*sqrt(3)/(sqrt(pi)*r^(3/2)))

        ::

            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: H = (1 - 2*x - y) * (1 - x - 2*y)
            sage: Hfac = H.factor()
            sage: G = exp(x + y)/Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: F
            (e^(x + y), [(x + 2*y - 1, 1), (2*x + y - 1, 1)])
            sage: p = {x: 1/3, y: 1/3}
            sage: alpha = (var('a'), var('b'))
            sage: F.asymptotics_multiple(p, alpha, 2, var('r')) # long time
            (3*((1/3)^(-a)*(1/3)^(-b))^r*e^(2/3), (1/3)^(-a)*(1/3)^(-b), 3*e^(2/3))
        """
        from itertools import product
        from sage.calculus.functions import jacobian
        from sage.calculus.var import function
        from sage.combinat.combinat import stirling_number1
        from sage.functions.log import exp, log
        from sage.functions.other import factorial, sqrt
        from sage.matrix.constructor import matrix
        from sage.misc.mrange import xmrange
        from sage.modules.free_module_element import vector
        from sage.rings.all import CC
        from sage.rings.arith import binomial
        from sage.rings.rational_field import QQ
        from sage.symbolic.constants import pi
        from sage.symbolic.relation import solve

        R = self.denominator_ring

        # Coerce keys of p into R.
        p = coerce_point(R, p)

        d = self.dimension()
        I = sqrt(-ZZ.one())
        # Coerce everything into the Symbolic Ring.
        X = [SR(x) for x in R.gens()]
        G = SR(self.numerator())
        H = [SR(h) for (h, e) in self.denominator_factored()]
        Hprod = prod(H)
        n = len(H)
        P = {SR(x): p[x] for x in R.gens()}
        Sstar = self._crit_cone_combo(p, alpha, coordinate)

        # Put the given convenient variable at end of variable list.
        if coordinate is not None:
            x = X.pop(coordinate)
            X.append(x)
            a = alpha.pop(coordinate)
            alpha.append(a)

        # Case n = d.
        if n == d:
            det = jacobian(H, X).subs(P).determinant().abs()
            exp_scale = prod([(P[X[i]] ** (-alpha[i])).subs(P)
                              for i in range(d)])
            subexp_part = G.subs(P) / (det * prod(P.values()))
            if numerical:
                exp_scale = exp_scale.n(digits=numerical)
                subexp_part = subexp_part.n(digits=numerical)
            return (exp_scale ** asy_var * subexp_part, exp_scale, subexp_part)

        # Case n < d.
        # If P is a tuple of rationals, then compute with it directly.
        # Otherwise, compute symbolically and plug in P at the end.
        if vector(P.values()) not in QQ ** d:
            sP = [var('p' + str(j)) for j in range(d)]
            P = {X[j]: sP[j] for j in range(d)}
            p = {sP[j]: p[X[j]] for j in range(d)}

        # Setup.
        if verbose:
            print("Creating auxiliary functions...")
        # Create T and S variables.
        t = 't'
        L = [str(elt) for elt in X]
        while t in L:
            t = t + 't'
        T = [var(t + str(i)) for i in range(d - 1)]
        s = 's'
        while s in L:
            s = s + 't'
        S = [var(s + str(i)) for i in range(n - 1)]
        Sstar = {S[j]: Sstar[j] for j in range(n - 1)}
        thetastar = {t: ZZ.zero() for t in T}
        thetastar.update(Sstar)
        # Create implicit functions.
        h = [function('h' + str(j))(*tuple(X[:d - 1])) for j in range(n)]
        U = function('U')(*tuple(X))
        # All other functions are defined in terms of h, U, and
        # explicit functions.
        Hcheck = prod([X[d - 1] - ZZ.one() / h[j] for j in range(n)])
        Gcheck = -G / U * prod([-h[j] / X[d - 1] for j in range(n)])
        A = [(-1) ** (n - 1) * X[d - 1] ** (-n + j) *
             diff(Gcheck.subs({X[d - 1]: ZZ.one() / X[d - 1]}), X[d - 1], j)
             for j in range(n)]
        e = {X[i]: P[X[i]] * exp(I * T[i]) for i in range(d - 1)}
        ht = [hh.subs(e) for hh in h]
        hsumt = (sum([S[j] * ht[j] for j in range(n - 1)]) +
                 (ZZ.one() - sum(S)) * ht[n - 1])
        At = [AA.subs(e).subs({X[d - 1]: hsumt}) for AA in A]
        Phit = (-log(P[X[d - 1]] * hsumt) +
                I * sum([alpha[i] / alpha[d - 1] * T[i]
                         for i in range(d - 1)]))
        # atP Stores h and U and all their derivatives evaluated at C.
        atP = P.copy()
        atP.update({hh.subs(P): ZZ.one() / P[X[d - 1]] for hh in h})

        # Compute the derivatives of h up to order 2 * N and evaluate at P.
        hderivs1 = {}   # First derivatives of h.
        for (i, j) in xmrange([d - 1, n], tuple):
            s = solve(diff(H[j].subs({X[d - 1]: ZZ.one() / h[j]}), X[i]),
                      diff(h[j], X[i]))[0].rhs().simplify()
            hderivs1.update({diff(h[j], X[i]): s})
            atP.update({diff(h[j], X[i]).subs(P): s.subs(P).subs(atP)})
        hderivs = diff_all(h, X[0:d - 1], 2 * N, sub=hderivs1, rekey=h)
        for k in hderivs:
            atP.update({k.subs(P): hderivs[k].subs(atP)})

        # Compute the derivatives of U up to order 2 * N - 2 + min{n, N} - 1 and
        # evaluate at P.
        # To do this, differentiate H = U*Hcheck over and over, evaluate at P,
        # and solve for the derivatives of U at P.
        # Need the derivatives of H with short keys to pass on to
        # diff_prod later.
        if verbose:
            print("Computing derivatives of auxiliary functions...")
        m = min(n, N)
        end = [X[d-1] for j in range(n)]
        Hprodderivs = diff_all(Hprod, X, 2 * N - 2 + n, ending=end, sub_final=P)
        atP.update({U.subs(P): diff(Hprod, X[d - 1], n).subs(P)/factorial(n)})
        Uderivs = {}
        k = Hprod.polynomial(CC).degree() - n
        if k == 0:
            # Then we can conclude that all higher derivatives of U are zero.
            for l in range(1, 2 * N - 2 + m):
                for s in combinations_with_replacement(X, l):
                    Uderivs[diff(U, list(s)).subs(P)] = ZZ.zero()
        elif k > 0 and k < 2 * N - 2 + m - 1:
            all_zero = True
            Uderivs = diff_prod(Hprodderivs, U, Hcheck, X,
                                range(1, k + 1), end, Uderivs, atP)
            # Check for a nonzero U derivative.
            if any(u for u in Uderivs.values()):
                all_zero = False
            if all_zero:
                # Then all higher derivatives of U are zero.
                for l in range(k + 1, 2 * N - 2 + m):
                    for s in combinations_with_replacement(X, l):
                        Uderivs.update({diff(U, list(s)).subs(P): ZZ.zero()})
            else:
                # Have to compute the rest of the derivatives.
                Uderivs = diff_prod(Hprodderivs, U, Hcheck, X,
                                    range(k + 1, 2 * N - 2 + m), end,
                                    Uderivs, atP)
        else:
            Uderivs = diff_prod(Hprodderivs, U, Hcheck, X,
                                range(1, 2 * N - 2 + m), end, Uderivs, atP)
        atP.update(Uderivs)
        Phit1 = jacobian(Phit, T + S).subs(hderivs1)
        a = jacobian(Phit1, T + S).subs(hderivs1).subs(thetastar).subs(atP)
        a_inv = a.inverse()
        Phitu = (Phit - (1 / Integer(2)) * matrix([T + S]) * a *
                 matrix([T + S]).transpose())
        Phitu = Phitu[0][0]

        # Compute all partial derivatives of At and Phitu up to orders 2 * N - 2
        # and 2 * N, respectively.  Take advantage of the fact that At and Phitu
        # are sufficiently differentiable functions so that mixed partials
        # are equal.  Thus only need to compute representative partials.
        # Choose nondecreasing sequences as representative differentiation-
        # order sequences.
        # To speed up later computations, create symbolic functions AA and BB
        # to stand in for the expressions At and Phitu respectively.
        if verbose:
            print("Computing derivatives of more auxiliary functions...")
        AA = [function('A' + str(j))(*tuple(T + S)) for j in range(n)]
        At_derivs = diff_all(At, T + S, 2 * N - 2, sub=hderivs1,
                             sub_final=[thetastar, atP], rekey=AA)
        BB = function('BB')(*tuple(T + S))
        Phitu_derivs = diff_all(Phitu, T + S, 2 * N, sub=hderivs1,
                                sub_final=[thetastar, atP], rekey=BB, zero_order=3)
        AABB_derivs = At_derivs
        AABB_derivs.update(Phitu_derivs)
        for j in range(n):
            AABB_derivs[AA[j]] = At[j].subs(thetastar).subs(atP)
        AABB_derivs[BB] = Phitu.subs(thetastar).subs(atP)

        if verbose:
            print("Computing second-order differential operator actions...")
        DD = diff_op(AA, BB, AABB_derivs, T + S, a_inv, n, N)
        L = {}
        for (j, k) in product(range(min(n, N)), range(max(0, N - 1 - n), N)):
            if j + k <= N - 1:
                L[(j, k)] = sum([DD[(j, k, l)] / ((-1) ** k * 2 ** (k + l) *
                                                  factorial(l) *
                                                  factorial(k + l))
                                 for l in range(2 * k + 1)])
                det = (a.determinant() ** (-1 / Integer(2)) *
                       (2 * pi) ** ((n - d) / Integer(2)))
        chunk = det * sum([(alpha[d - 1] * asy_var) ** ((n - d) /
                                                        Integer(2) - q) *
                           sum([L[(j, k)] * binomial(n - 1, j) *
                                stirling_number1(n - j, n + k - q) *
                                (-1) ** (q - j - k)
                                for (j, k) in product(range(min(n - 1, q) + 1),
                                                      range(max(0, q - n),
                                                             q + 1))
                                if j + k <= q])
                           for q in range(N)])
        chunk = chunk.subs(P).simplify()
        coeffs = chunk.coefficients(asy_var)
        coeffs.reverse()
        coeffs = coeffs[:N]
        if numerical:
            subexp_part = sum([co[0].subs(p).n(digits=numerical) *
                               asy_var ** co[1]
                               for co in coeffs])
            exp_scale = prod([(P[X[i]] ** (-alpha[i])).subs(p)
                              for i in range(d)]).n(digits=numerical)
        else:
            subexp_part = sum([co[0].subs(p) * asy_var ** co[1]
                               for co in coeffs])
            exp_scale = prod([(P[X[i]] ** (-alpha[i])).subs(p)
                              for i in range(d)])
        return (exp_scale ** asy_var * subexp_part, exp_scale, subexp_part)

    def _crit_cone_combo(self, p, alpha, coordinate=None):
        r"""
        Return an auxiliary point associated to the multiple
        point ``p`` of the factors ``self``.

        INPUT:

        - ``p`` -- a dictionary with keys that can be coerced to equal
          ``self.denominator_ring.gens()``
        - ``alpha`` -- a list of rationals

        OUTPUT:

        A solution of the matrix equation `y \Gamma = \alpha^{\prime}` for `y`,
        where `\Gamma` is the matrix given by
        ``[direction(v) for v in self.log_grads(p)]`` and
        `\alpha^{\prime}` is ``direction(alpha)``.

        .. SEEALSO::

            :func:`direction`

        .. NOTE::

            For internal use by
            :meth:`FractionWithFactoredDenominator.asymptotics_multiple()`.

        .. NOTE::

            Use this function only when `\Gamma` is well-defined and
            there is a unique solution to the matrix equation
            `y \Gamma = \alpha'`. Fails otherwise.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: p = exp(x)
            sage: df = [(1 - 2*x - y, 1), (1 - x - 2*y, 1)]
            sage: f = FFPD(p, df)
            sage: p = {x: 1/3, y: 1/3}
            sage: alpha = (var('a'), var('b'))
            sage: f._crit_cone_combo(p, alpha)
            [1/3*(2*a - b)/b, -2/3*(a - 2*b)/b]
        """
        from sage.matrix.constructor import matrix
        from sage.symbolic.relation import solve

        # Assuming here that each log_grads(f) has nonzero final component.
        # Then 'direction' will not throw a division by zero error.
        R = self.denominator_ring

        # Coerce keys of p into R.
        p = coerce_point(R, p)

        d = self.dimension()
        n = len(self.denominator_factored())
        Gamma = matrix([direction(v, coordinate) for v in self.log_grads(p)])
        beta = direction(alpha, coordinate)
        # solve_left() fails when working in SR :-(.
        # So use solve() instead.
        # Gamma.solve_left(vector(beta))
        V = [var('sss' + str(i)) for i in range(n)]
        M = matrix(V) * Gamma
        eqns = [M[0][j] == beta[j] for j in range(d)]
        s = solve(eqns, V, solution_dict=True)[0]  # Assume a unique solution.
        return [s[v] for v in V]

    def grads(self, p):
        r"""
        Return a list of the gradients of the polynomials
        ``[q for (q, e) in self.denominator_factored()]`` evalutated at ``p``.

        INPUT:

        - ``p`` -- (optional; default: ``None``) a dictionary whose keys are
          the generators of ``self.denominator_ring``

        OUTPUT:

        A list.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: p = exp(x)
            sage: df = [(x^3 + 3*y^2, 5), (x*y, 2), (y, 1)]
            sage: f = FFPD(p, df)
            sage: f
            (e^x, [(y, 1), (x*y, 2), (x^3 + 3*y^2, 5)])
            sage: R.gens()
            (x, y)
            sage: p = None
            sage: f.grads(p)
            [(0, 1), (y, x), (3*x^2, 6*y)]

            sage: p = {x: sqrt(2), y: var('a')}
            sage: f.grads(p)
            [(0, 1), (a, sqrt(2)), (6, 6*a)]
        """
        R = self.denominator_ring

        # Coerce keys of p into R.
        p = coerce_point(R, p)

        X = R.gens()
        d = self.dimension()
        H = [h for (h, e) in self.denominator_factored()]
        n = len(H)
        return [tuple([diff(H[i], X[j]).subs(p) for j in range(d)])
                for i in range(n)]

    def log_grads(self, p):
        r"""
        Return a list of the logarithmic gradients of the polynomials
        ``[q for (q, e) in self.denominator_factored()]`` evalutated at ``p``.

        The logarithmic gradient of a function `f` at point `p` is the
        vector `(x_1 \partial_1 f(x), \ldots, x_d \partial_d f(x) )`
        evaluated at `p`.

        INPUT:

        - ``p`` -- (optional; default: ``None``) a dictionary whose keys
          are the generators of ``self.denominator_ring``

        OUTPUT:

        A list.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: p = exp(x)
            sage: df = [(x^3 + 3*y^2, 5), (x*y, 2), (y, 1)]
            sage: f = FFPD(p, df)
            sage: f
            (e^x, [(y, 1), (x*y, 2), (x^3 + 3*y^2, 5)])
            sage: R.gens()
            (x, y)
            sage: p = None
            sage: f.log_grads(p)
            [(0, y), (x*y, x*y), (3*x^3, 6*y^2)]

            sage: p = {x: sqrt(2), y: var('a')}
            sage: f.log_grads(p)
            [(0, a), (sqrt(2)*a, sqrt(2)*a), (6*sqrt(2), 6*a^2)]
        """
        R = self.denominator_ring

        # Coerce keys of p into R.
        p = coerce_point(R, p)

        X = R.gens()
        d = self.dimension()
        H = [h for (h, e) in self.denominator_factored()]
        n = len(H)
        return [tuple([(X[j] * diff(H[i], X[j])).subs(p) for j in range(d)])
                for i in range(n)]

    def critical_cone(self, p, coordinate=None):
        r"""
        Return the critical cone of the convenient multiple point ``p``.

        INPUT:

        - ``p`` -- a dictionary with keys that can be coerced to equal
          ``self.denominator_ring.gens()`` and values in a field
        - ``coordinate`` -- (optional; default: ``None``) a natural number

        OUTPUT:

        A list of vectors.

        This list of vectors generate the critical cone of ``p`` and
        the cone itself, which is ``None`` if the values of ``p`` don't lie in
        `\QQ`. Divide logarithmic gradients by their component ``coordinate``
        entries. If ``coordinate = None``, then search from `d-1` down to 0
        for the first index ``j`` such that for all ``i`` we have
        ``self.log_grads()[i][j] != 0`` and set ``coordinate = j``.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y,z> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: G = 1
            sage: H = (1 - x*(1 + y)) * (1 - z*x**2*(1 + 2*y))
            sage: Hfac = H.factor()
            sage: G = 1/Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: p = {x: 1/2, y: 1, z: 4/3}
            sage: F.critical_cone(p)
            ([(2, 1, 0), (3, 1, 3/2)], 2-d cone in 3-d lattice N)
        """
        from sage.geometry.cone import Cone

        R = self.denominator_ring

        # Coerce keys of p into R.
        p = coerce_point(R, p)

        d = self.dimension()
        lg = self.log_grads(p)
        n = len(lg)
        if coordinate not in range(d):
            # Search from d-1 down to 0 for a coordinate j such that
            # for all i we have lg[i][j] != 0.
            # One is guaranteed to exist in the case of a convenient multiple
            # point.
            for j in reversed(range(d)):
                if 0 not in [lg[i][j] for i in range(n)]:
                    coordinate = j
                    break
        Gamma = [direction(v, coordinate) for v in lg]
        try:
            cone = Cone(Gamma)
        except TypeError:
            cone = None
        return (Gamma, cone)

    def is_convenient_multiple_point(self, p):
        r"""
        Tests if ``p`` is a convenient multiple point of ``self``.

        In case ``p`` is a convenient multiple point, ``verdict = True`` and
        ``comment`` is a string stating which variables it's convenient to use.
        In case ``p`` is not, ``verdict = False`` and ``comment`` is a string
        explaining why ``p`` fails to be a convenient multiple point.

        See [RaWi2012]_ for more details.

        INPUT:

        - ``p`` -- a dictionary with keys that can be coerced to equal
          ``self.denominator_ring.gens()``

        OUTPUT:

        A pair ``(verdict, comment)``.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y,z> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: H = (1 - x*(1 + y)) * (1 - z*x**2*(1 + 2*y))
            sage: df = H.factor()
            sage: G = 1 / df.unit()
            sage: F = FFPD(G, df)
            sage: p1 = {x: 1/2, y: 1, z: 4/3}
            sage: p2 = {x: 1, y: 2, z: 1/2}
            sage: F.is_convenient_multiple_point(p1)
            (True, 'convenient in variables [x, y]')
            sage: F.is_convenient_multiple_point(p2)
            (False, 'not a singular point')
        """
        from sage.combinat.subset import Subsets
        from sage.matrix.constructor import matrix

        R = self.denominator_ring

        # Coerce keys of p into R.
        p = coerce_point(R, p)

        H = [h for (h, e) in self.denominator_factored()]
        n = len(H)
        d = self.dimension()

        # Test 1: Are the factors in H zero at p?
        if [h.subs(p) for h in H] != [0 for h in H]:
            # Failed test 1.  Move on to next point.
            return (False, 'not a singular point')

        # Test 2: Are the factors in H smooth at p?
        grads = self.grads(p)
        for v in grads:
            if v == [0 for i in range(d)]:
                return (False, 'not smooth point of factors')

        # Test 3: Do the factors in H intersect transversely at p?
        if n <= d:
            M = matrix(grads)
            if M.rank() != n:
                return (False, 'not a transverse intersection')
        else:
            # Check all sub-multisets of grads of size d.
            for S in Subsets(grads, d, submultiset=True):
                M = matrix(S)
                if M.rank() != d:
                    return (False, 'not a transverse intersection')

        # Test 4: Is p convenient?
        M = matrix(self.log_grads(p))
        convenient_coordinates = []
        for j in range(d):
            if 0 not in M.columns()[j]:
                convenient_coordinates.append(j)
        if not convenient_coordinates:
            return (False, 'multiple point but not convenient')

        # Tests all passed
        X = R.gens()
        convenientX = [X[i] for i in convenient_coordinates]
        return (True, 'convenient in variables {}'.format(convenientX))

    def singular_ideal(self):
        r"""
        Return the singular ideal of ``self``.

        Let `R` be the ring of ``self`` and `H` its denominator.
        Let `H_{red}` be the reduction (square-free part) of `H`.
        Return the ideal in `R` generated by `H_{red}` and
        its partial derivatives.
        If the coefficient field of `R` is algebraically closed,
        then the output is the ideal of the singular locus (which
        is a variety) of the variety of `H`.

        OUTPUT:

        An ideal.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y,z> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: H = (1 - x*(1 + y))^3 * (1 - z*x**2*(1 + 2*y))
            sage: df = H.factor()
            sage: G = 1 / df.unit()
            sage: F = FFPD(G, df)
            sage: F.singular_ideal()
            Ideal (x*y + x - 1, y^2 - 2*y*z + 2*y - z + 1, x*z + y - 2*z + 1) of
             Multivariate Polynomial Ring in x, y, z over Rational Field
        """
        R = self.denominator_ring

        Hred = prod([h for (h, e) in self.denominator_factored()])
        J = R.ideal([Hred] + Hred.gradient())
        return R.ideal(J.groebner_basis())

    def smooth_critical_ideal(self, alpha):
        r"""
        Return the smooth critical ideal of ``self``.

        Let `R` be the ring of ``self`` and `H` its denominator.
        Return the ideal in `R` of smooth critical points of the variety
        of `H` for the direction ``alpha``.
        If the variety `V` of `H` has no smooth points, then return the ideal
        in `R` of `V`.

        See [RaWi2012]_ for more details.

        INPUT:

        - ``alpha`` -- a tuple of positive integers and/or symbolic entries
          of length ``self.denominator_ring.ngens()``

        OUTPUT:

        An ideal.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: H = (1 - x - y - x*y)^2
            sage: Hfac = H.factor()
            sage: G = 1/Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: alpha = var('a1, a2')
            sage: F.smooth_critical_ideal(alpha)
            Ideal (y^2 + 2*a1/a2*y - 1, x + ((-a2)/a1)*y + (-a1 + a2)/a1) of
             Multivariate Polynomial Ring in x, y over Fraction Field of
             Multivariate Polynomial Ring in a1, a2 over Rational Field

            sage: H = (1-x-y-x*y)^2
            sage: Hfac = H.factor()
            sage: G = 1/Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: alpha = [7/3, var('a')]
            sage: F.smooth_critical_ideal(alpha)
            Ideal (y^2 + 14/(3*a)*y - 1, x + (-3/7*a)*y + 3/7*a - 1) of
             Multivariate Polynomial Ring in x, y over Fraction Field of
             Univariate Polynomial Ring in a over Rational Field
        """
        from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing

        R = self.denominator_ring
        Hred = prod([h for (h, e) in self.denominator_factored()])
        K = R.base_ring()
        d = self.dimension()

        # Expand K by the variables of alpha if there are any.
        indets = []
        for a in alpha:
            if a not in K and a in SR:
                indets.append(a)
        indets = sorted(set(indets), key=str)   # Delete duplicates in indets.
        if indets:
            L = PolynomialRing(K, indets).fraction_field()
            S = R.change_ring(L)
            # Coerce alpha into L.
            alpha = [L(a) for a in alpha]
        else:
            S = R

        # Find smooth, critical points for alpha.
        X = S.gens()
        Hred = S(Hred)
        J = S.ideal([Hred] +
                    [alpha[d - 1] * X[i] * diff(Hred, X[i]) -
                     alpha[i] * X[d - 1] * diff(Hred, X[d - 1])
                     for i in range(d - 1)])
        return S.ideal(J.groebner_basis())

    def maclaurin_coefficients(self, multi_indices, numerical=0):
        r"""
        Return the Maclaurin coefficients of ``self`` with given
        ``multi_indices``.

        INPUT:

        - ``multi_indices`` -- a list of tuples of positive integers, where
          each tuple has length ``self.dimension()``
        - ``numerical`` -- (optional; default: 0) a natural number; if
          positive, return numerical approximations of coefficients with
          ``numerical`` digits of accuracy

        OUTPUT:

        A dictionary whose value of the key ``nu`` are the Maclaurin
        coefficient of index ``nu`` of ``self``.

        .. NOTE::

            Uses iterated univariate Maclaurin expansions. Slow.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: H = 2 - 3*x
            sage: Hfac = H.factor()
            sage: G = 1 / Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: F
            (-1/3, [(x - 2/3, 1)])
            sage: F.maclaurin_coefficients([(2*k,) for k in range(6)])
            {(0,): 1/2,
             (2,): 9/8,
             (4,): 81/32,
             (6,): 729/128,
             (8,): 6561/512,
             (10,): 59049/2048}

        ::

            sage: R.<x,y,z> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: H = (4 - 2*x - y - z) * (4 - x - 2*y - z)
            sage: Hfac = H.factor()
            sage: G = 16 / Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: alpha = vector([3, 3, 2])
            sage: interval = [1, 2, 4]
            sage: S = [r*alpha for r in interval]
            sage: F.maclaurin_coefficients(S, numerical=10)
            {(3, 3, 2): 0.7849731445,
             (6, 6, 4): 0.7005249476,
             (12, 12, 8): 0.5847732654}
        """
        R = self.denominator_ring
        d = self.dimension()
        coeffs = {}

        # Deal with the simple univariate case first.
        if d == 1:
            f = SR(self.quotient())
            x = SR(R.gens()[0])
            m = max(multi_indices)[0]
            f = f.taylor(x, 0, m)
            F = R(f)
            tmp = F.coefficients()
            for nu in multi_indices:
                val = tmp[nu[0]]
                if numerical:
                    val = val.n(digits=numerical)
                coeffs[tuple(nu)] = val
            return coeffs

        # Create biggest multi-index needed.
        alpha = []
        for i in range(d):
            alpha.append(max(nu[i] for nu in multi_indices))

        # Compute Maclaurin expansion of self up to index alpha.
        # Use iterated univariate expansions.
        # Slow!
        f = SR(self.quotient())
        X = [SR(g) for g in R.gens()]
        for i in range(d):
            f = f.taylor(X[i], 0, alpha[i])
        F = R(f)

        # Collect coefficients.
        X = R.gens()
        for nu in multi_indices:
            monomial = prod(X[i] ** nu[i] for i in range(d))
            val = F.monomial_coefficient(monomial)
            if numerical:
                val = val.n(digits=numerical)
            coeffs[tuple(nu)] = val
        return coeffs

    def relative_error(self, approx, alpha, interval, exp_scale=Integer(1),
                       digits=10):
        r"""
        Return the relative error between the values of the Maclaurin
        coefficients of ``self`` with multi-indices ``r alpha`` for ``r`` in
        ``interval`` and the values of the functions (of the variable ``r``)
        in ``approx``.

        INPUT:

        - ``approx`` -- an individual or list of symbolic expressions in
          one variable
        - ``alpha`` - a list of positive integers of length
          ``self.denominator_ring.ngens()``
        - ``interval`` -- a list of positive integers
        - ``exp_scale`` -- (optional; default: 1) a number

        OUTPUT:

        A list of tuples with properties described below.

        This outputs a list whose entries are a tuple
        ``(r*alpha, a_r, b_r, err_r)`` for ``r`` in ``interval``.
        Here ``r*alpha`` is a tuple; ``a_r`` is the ``r*alpha`` (multi-index)
        coefficient of the Maclaurin series for ``self`` divided by
        ``exp_scale**r``;
        ``b_r`` is a list of the values of the functions in ``approx``
        evaluated at ``r`` and divided by ``exp_scale**m``;
        ``err_r`` is the list of relative errors
        ``(a_r - f)/a_r`` for ``f`` in ``b_r``.
        All outputs are decimal approximations.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: H = 1 - x - y - x*y
            sage: Hfac = H.factor()
            sage: G = 1 / Hfac.unit()
            sage: F = FFPD(G, Hfac)
            sage: alpha = [1, 1]
            sage: r = var('r')
            sage: a1 = (0.573/sqrt(r))*5.83^r
            sage: a2 = (0.573/sqrt(r) - 0.0674/r^(3/2))*5.83^r
            sage: es = 5.83
            sage: F.relative_error([a1, a2], alpha, [1, 2, 4, 8], es) # long time
            [((1, 1), 0.5145797599,
              [0.5730000000, 0.5056000000], [-0.1135300000, 0.01745066667]),
             ((2, 2), 0.3824778089,
              [0.4051721856, 0.3813426871], [-0.05933514614, 0.002967810973]),
             ((4, 4), 0.2778630595,
              [0.2865000000, 0.2780750000], [-0.03108344267, -0.0007627515584]),
             ((8, 8), 0.1991088276,
              [0.2025860928, 0.1996074055], [-0.01746414394, -0.002504047242])]
        """
        from sage.modules.free_module_element import vector

        if not isinstance(approx, (list, tuple)):
            approx = [approx]
        if approx[0].variables():
            av = approx[0].variables()[0]
        else:
            av = ZZ.one()

        # Get Maclaurin coefficients of self.
        alpha = vector(alpha)
        multi_indices = [r * alpha for r in interval]
        mac = self.maclaurin_coefficients(multi_indices, numerical=digits)
        #mac = self.old_maclaurin_coefficients(alpha, max(interval))
        mac_approx = {}
        stats = []
        for r in interval:
            exp_s_r = exp_scale ** r
            beta = tuple(r * alpha)
            mac[beta] = (mac[beta] / exp_s_r).n(digits=digits)
            mac_approx[beta] = [(f.subs({av: r}) / exp_s_r).n(digits=digits)
                                for f in approx]
            stats_row = [beta, mac[beta], mac_approx[beta]]
            if mac[beta] == 0:
                stats_row.extend([None for a in mac_approx[beta]])
            else:
                stats_row.append([(mac[beta] - a) / mac[beta]
                                  for a in mac_approx[beta]])
            stats.append(tuple(stats_row))
        return stats

    def _add_(left, right):
        r"""
        Returns the sum of ``left`` with ``right``.

        INPUT:

        - ``left`` -- the left summand (i.e. ``self``)

        - ``right`` -- the right summand

        OUTPUT:

        The sum as a new element.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: df = (x, 1), (y, 1), (x*y + 1, 1)
            sage: f = FFPD(2, df)
            sage: g = FFPD(2*x*y, df)
            sage: f + g
            (2, [(y, 1), (x, 1)])
        """
        return FractionWithFactoredDenominatorSum([left, right]).sum()

    def _mul_(left, right):
        r"""
        Returns the product of ``left`` with ``right``.

        INPUT:

        - ``left`` -- the left factor (i.e. ``self``)

        - ``right`` -- the right factor

        OUTPUT:

        The product as a new element.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: f = FFPD(2, [(x, 1), (x*y + 1, 1), (x*y^2 + 1, 1)])
            sage: g = FFPD(2*x*y, [(y, 1), (x*y + 1, 1), (x^2*y + 1, 1)])
            sage: f * g
            (4, [(x*y + 1, 1), (x*y + 1, 1), (x*y^2 + 1, 1), (x^2*y + 1, 1)])
        """
        numer = left.numerator() * right.numerator()
        df = left.denominator_factored() + right.denominator_factored()
        return left.parent()(numer, df)


class FractionWithFactoredDenominatorRing(UniqueRepresentation, Ring):
    r"""
    This is the ring of fractions with factored denominator.

    INPUT:

    - ``denominator_ring`` -- the base ring (a polynomial ring)

    - ``numerator_ring`` -- (optional) the numerator ring; the default is
      the ``denominator_ring``

    - ``category`` -- (default: :class:`Rings`) the category

    .. SEEALSO::

        :class:`FractionWithFactoredDenominator`,
        :mod:`~sage.rings.asymptotic.asymptotics_multivariate_generating_functions`

    EXAMPLES::

        sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
        sage: R.<x,y> = PolynomialRing(QQ)
        sage: FFPD = FractionWithFactoredDenominatorRing(R)
        sage: df = [x, 1], [y, 1], [x*y+1, 1]
        sage: f = FFPD(x, df)  # indirect doctest
        sage: f
        (1, [(y, 1), (x*y + 1, 1)])

    AUTHORS:

    - Daniel Krenn (2014-12-01)
    """

    @staticmethod
    def __classcall_private__(cls, denominator_ring, numerator_ring=None, category=None):
        """
        Normalize input to ensure a unique representation.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD1 = FractionWithFactoredDenominatorRing(R)
            sage: FFPD2 = FractionWithFactoredDenominatorRing(R, R, Rings())
            sage: FFPD1 is FFPD2
            True
        """
        if numerator_ring is None:
            numerator_ring = denominator_ring
        if not numerator_ring.has_coerce_map_from(denominator_ring):
            raise ValueError('numerator ring {} has no coercion map from the '
                             'denominator ring {}'.format(
                                 numerator_ring, denominator_ring))
        category = Rings().or_subcategory(category)
        return super(FractionWithFactoredDenominatorRing, cls).__classcall__(cls,
                        denominator_ring, numerator_ring, category)

    def __init__(self, denominator_ring, numerator_ring=None, category=None):
        r"""
        Initialize ``self``.

        TESTS::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: P.<X, Y> = ZZ[]
            sage: FractionWithFactoredDenominatorRing(P)
            Ring of fractions with factored denominator
            over Multivariate Polynomial Ring in X, Y over Integer Ring
        """
        self._numerator_ring = numerator_ring
        self._denominator_ring = denominator_ring
        Ring.__init__(self, denominator_ring, category=category)

    def _repr_(self):
        r"""
        Returns a representation.

        OUTPUT:

        A string.

        TESTS::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: P.<X, Y> = ZZ[]
            sage: FractionWithFactoredDenominatorRing(P)  # indirect doctest
            Ring of fractions with factored denominator
            over Multivariate Polynomial Ring in X, Y over Integer Ring
        """
        return ("Ring of fractions with factored denominator "
                "over {!r}".format(self.base()))

    def base_ring(self):
        r"""
        Returns the base ring.

        OUTPUT:

        A ring.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: P.<X, Y> = ZZ[]
            sage: F = FractionWithFactoredDenominatorRing(P); F
            Ring of fractions with factored denominator
            over Multivariate Polynomial Ring in X, Y over Integer Ring
            sage: F.base_ring()
            Integer Ring
            sage: F.base()
            Multivariate Polynomial Ring in X, Y over Integer Ring
        """
        return self.base().base_ring()

    from sage.misc.decorators import rename_keyword
    @rename_keyword(deprecation=10519, reduce_='reduce')
    def _element_constructor_(self, *args, **kwargs):
        r"""
        Returns an element of this ring.

        See :class:`FractionWithFactoredDenominator` for details.

        TESTS::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: df = [x, 1], [y, 1], [x*y+1, 1]
            sage: f = FFPD(x, df)  # indirect doctest
            sage: f
            (1, [(y, 1), (x*y + 1, 1)])
        """
        R = self.base()
        Q = R.fraction_field()

        # process deprecated keyword arguments
        hasn = 'numerator' in kwargs
        hasdf = 'denominator_factored' in kwargs
        if hasn:
            from sage.misc.superseded import deprecation
            deprecation(10519, "Keyword argument 'numerator' "
                               "is deprecated. "
                               "Ignoring non-keyword arguments (if any). "
                               "Specify numerator and factored denominator "
                               "as first and second argument, i.e., use "
                               "something like FFPD(n, df).")
        if hasdf:
            from sage.misc.superseded import deprecation
            deprecation(10519, "Keyword argument 'denominator_factored' "
                               "is deprecated. "
                               "Ignoring non-keyword arguments (if any). "
                               "Specify numerator and factored denominator "
                               "as first and second argument, i.e., use "
                               "something like FFPD(n, df).")
        if hasn or hasdf:
            args = [kwargs.pop('numerator') if hasn else R(0),
                    kwargs.pop('denominator_factored') if hasdf else []]

        hasq = 'quotient' in kwargs
        if hasq:
            from sage.misc.superseded import deprecation
            deprecation(10519, "Keyword argument 'quotient' "
                               "is deprecated. "
                               "Ignoring non-keyword arguments (if any). "
                               "Specify numerator and factored denominator "
                               "as first and second argument, i.e., use "
                               "something like FFPD(q).")
            args = [kwargs.pop('quotient')]

        if (hasn or hasdf) and hasq:
            raise ValueError('parameters ambiguous')

        # process keyword arguments
        reduce = kwargs.pop('reduce', None)

        if kwargs:
            raise ValueError('Unknown keyword arguments '
                             '%s given' % (kwargs,))

        # process arguments
        if len(args) > 2:
            raise ValueError('too many arguments given')

        elif not args:
            raise ValueError('No argument given. '
                             'We are in serious troubles...')

        # At this point we have one or two input arguments.

        x = args[0]
        try:
            P = x.parent()
        except AttributeError:
            P = None

        denominator_factored = None  # init
        reduce_default = True

        if len(args) == 2:
            numerator, denominator_factored = args
            if numerator is None:
                numerator = R(0)
            if denominator_factored is None:
                denominator_factored = []

            from sage.rings.semirings.non_negative_integer_semiring import NN
            try:
                denominator_factored = sorted(
                    (R(d[0]), NN(d[1])) for d in denominator_factored)
            except TypeError:
                raise TypeError('factored denominator is not well-formed '
                                'or of wrong type')

        # From now on we only have one input argument;
        # it's called x and has parent P.

        elif isinstance(P, FractionWithFactoredDenominatorRing):
            numerator = x._numerator
            denominator_factored = self._denominator_factored
            reduce_default = False

        elif P == SR:
            numerator = x.numerator()
            denominator = x.denominator()
            reduce_default = False

        elif x in R:
            numerator = R(x)
            denominator_factored = []

        elif x in Q:
            quotient = Q(x)
            numerator = quotient.numerator()
            denominator = quotient.denominator()
            reduce_default = False

        elif hasattr(x, 'numerator') and hasattr(x, 'denominator'):
            numerator = x.numerator()
            denominator = x.denominator()
            reduce_default = False

        else:
            raise TypeError('element {} is not contained in {}'.format(x, self))

        if reduce is None:
            reduce = reduce_default

        if denominator_factored is None:
            if denominator not in R:
                raise TypeError('extracted denominator {} is not in {}'.format(denominator, self))
            p = numerator
            q = R(denominator)

            from sage.rings.polynomial.polynomial_ring import is_PolynomialRing
            from sage.rings.polynomial.multi_polynomial_ring_generic import is_MPolynomialRing
            if is_PolynomialRing(R) or is_MPolynomialRing(R):
                if not R(q).is_unit():
                    # Factor denominator
                    try:
                        df = q.factor()
                    except NotImplementedError:
                        # Singular's factor() needs 'proof=False'.
                        df = q.factor(proof=False)
                    numerator = p / df.unit()
                    df = sorted(tuple(t) for t in df)  # sort for consistency
                    denominator_factored = df
            else:
                # At this point, denominator could not be factored.
                numerator = p / q
                denominator_factored = []

        return self.element_class(self,
                                  numerator=numerator,
                                  denominator_factored=denominator_factored,
                                  reduce=reduce)

    def _coerce_map_from_(self, P):
        r"""
        Checks if there is a coercion from the given parent.

        INPUT:

        - ``P`` -- a parent

        OUTPUT:

        ``True`` if there is a coercion, otherwise ``False`` or ``None``.

        TESTS::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: Q = QQ['x,y']
            sage: FFPD_QQ = FractionWithFactoredDenominatorRing(Q)
            sage: FFPD_QQ.has_coerce_map_from(Q)
            True
            sage: FFPD_QQ.has_coerce_map_from(QQ)
            True
            sage: FFPD_QQ.has_coerce_map_from(ZZ)
            True
            sage: FFPD_QQ.has_coerce_map_from(Q.fraction_field())
            True
            sage: Z = ZZ['x,y']
            sage: FFPD_ZZ = FractionWithFactoredDenominatorRing(Z)
            sage: FFPD_ZZ.has_coerce_map_from(FFPD_QQ)
            False
            sage: FFPD_QQ.has_coerce_map_from(FFPD_ZZ)
            True
            sage: FFPD_ZZ.has_coerce_map_from(QQ)
            False
            sage: FFPD_ZZ.has_coerce_map_from(Z.fraction_field())
            True
            sage: FFPD_ZZ.has_coerce_map_from(Q.fraction_field())
            False
            sage: FFPD_QQ.has_coerce_map_from(Z.fraction_field())
            True
        """
        if isinstance(P, FractionWithFactoredDenominatorRing):
            if self.base().has_coerce_map_from(P.base()):
                return True

        from sage.rings.fraction_field import is_FractionField
        if is_FractionField(P):
            B = P.base()
            from sage.rings.polynomial.polynomial_ring import is_PolynomialRing
            from sage.rings.polynomial.multi_polynomial_ring_generic import is_MPolynomialRing
            if is_PolynomialRing(B) or is_MPolynomialRing(B):
                if self.base().has_coerce_map_from(B):
                    return True

        if self.base().has_coerce_map_from(P):
            return True

    def _an_element_(self):
        r"""
        Returns an element.

        OUTPUT:

        An element.

        TESTS::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: FFPD.an_element()  # indirect doctest
            (42, [(x, 3)])
        """
        from sage.rings.semirings.non_negative_integer_semiring import NN
        return self(NN.an_element(), [(self.base().an_element(), NN(3))])


    Element = FractionWithFactoredDenominator


class FractionWithFactoredDenominatorSum(list):
    r"""
    A list representing the sum of :class:`FractionWithFactoredDenominator`
    objects with distinct denominator factorizations.

    AUTHORS:

    - Alexander Raichev (2012-06-25)

    - Daniel Krenn (2014-12-01)
    """

    def __repr__(self):
        r"""
        Return a string representation of ``self``.

        OUTPUT:

        A string.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing, FractionWithFactoredDenominatorSum
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: f = FFPD(x + y, [(y, 1), (x, 1)])
            sage: g = FFPD(x**2 + y, [(y, 1), (x, 2)])
            sage: FractionWithFactoredDenominatorSum([f, g])
            (x + y, [(y, 1), (x, 1)]) + (x^2 + y, [(y, 1), (x, 2)])
        """
        return ' + '.join(repr(r) for r in self)

    def __eq__(self, other):
        r"""
        Return ``True`` if ``self`` is equal to ``other``.

        OUTPUT:

        A boolean.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing, FractionWithFactoredDenominatorSum
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: f = FFPD(x + y, [(y, 1), (x, 1)])
            sage: g = FFPD(x*(x + y), [(y, 1), (x, 2)])
            sage: s = FractionWithFactoredDenominatorSum([f]); s
            (x + y, [(y, 1), (x, 1)])
            sage: t = FractionWithFactoredDenominatorSum([g]); t
            (x + y, [(y, 1), (x, 1)])
            sage: s == t
            True
        """
        from operator import methodcaller
        return (sorted(self, key=methodcaller('_total_order_key_')) ==
                sorted(other, key=methodcaller('_total_order_key_')))

    def __ne__(self, other):
        r"""
        Return ``True`` if ``self`` is not equal to ``other``.

        OUTPUT:

        A boolean.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing, FractionWithFactoredDenominatorSum
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: f = FFPD(x + y, [(y, 1), (x, 1)])
            sage: g = FFPD(x + y, [(y, 1), (x, 2)])
            sage: s = FractionWithFactoredDenominatorSum([f]); s
            (x + y, [(y, 1), (x, 1)])
            sage: t = FractionWithFactoredDenominatorSum([g]); t
            (x + y, [(y, 1), (x, 2)])
            sage: s != t
            True
        """
        return not self.__eq__(other)

    @property
    def denominator_ring(self):
        r"""
        Return the polynomial ring of the denominators of ``self``.

        OUTPUT:

        A ring or ``None`` if the list is empty.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing, FractionWithFactoredDenominatorSum
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R)
            sage: f = FFPD(x + y, [(y, 1), (x, 1)])
            sage: s = FractionWithFactoredDenominatorSum([f])
            sage: s.denominator_ring
            Multivariate Polynomial Ring in x, y over Rational Field
            sage: g = FFPD(x + y, [])
            sage: t = FractionWithFactoredDenominatorSum([g])
            sage: t.denominator_ring
            Multivariate Polynomial Ring in x, y over Rational Field
        """
        for r in self:
            return r.denominator_ring
        return None

    def whole_and_parts(self):
        r"""
        Rewrite ``self`` as a sum of a (possibly zero) polynomial
        followed by reduced rational expressions.

        OUTPUT:

        An instance of :class:`FractionWithFactoredDenominatorSum`.

        Only useful for multivariate decompositions.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing, FractionWithFactoredDenominatorSum
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: f = x**2 + 3*y + 1/x + 1/y
            sage: f = FFPD(f); f
            (x^3*y + 3*x*y^2 + x + y, [(y, 1), (x, 1)])
            sage: FractionWithFactoredDenominatorSum([f]).whole_and_parts()
            (x^2 + 3*y, []) + (x + y, [(y, 1), (x, 1)])

            sage: f = cos(x)**2 + 3*y + 1/x + 1/y; f
            cos(x)^2 + 3*y + 1/x + 1/y
            sage: G = f.numerator()
            sage: H = R(f.denominator())
            sage: f = FFPD(G, H.factor()); f
            (x*y*cos(x)^2 + 3*x*y^2 + x + y, [(y, 1), (x, 1)])
            sage: FractionWithFactoredDenominatorSum([f]).whole_and_parts()
            (0, []) + (x*y*cos(x)^2 + 3*x*y^2 + x + y, [(y, 1), (x, 1)])
        """
        whole = 0
        parts = []
        R = self.denominator_ring
        for r in self:
            # Since r has already passed through FFPD.__init__()'s reducing
            # procedure, r is already in lowest terms.
            # Check if can write r as a mixed fraction: whole + fraction.
            p = r.numerator()
            q = r.denominator()
            if q == 1:
                # r is already whole
                whole += p
            else:
                try:
                    # Coerce p into R and divide p by q
                    p = R(p)
                    a, b = p.quo_rem(q)
                except TypeError:
                    # p is not in R and so can't divide p by q
                    a = 0
                    b = p
                whole += a
                parts.append(r.parent()(b, r.denominator_factored(), reduce=False))
        return FractionWithFactoredDenominatorSum(
            [r.parent()(whole, ())] + parts)  # r.parent() is not the nicest here

    def _combine_like_terms_(self):
        r"""
        Combine terms in ``self`` with the same denominator.
        Only useful for multivariate decompositions.

        OUTPUT:

        An instance of :class:`FractionWithFactoredDenominatorSum`.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing, FractionWithFactoredDenominatorSum
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: f = FFPD(1/(x * y * (x*y + 1)))
            sage: g = FFPD(x/(x * y * (x*y + 1)))
            sage: s = FractionWithFactoredDenominatorSum([f, g, f])
            sage: t = s._combine_like_terms_()
            sage: s
            (1, [(y, 1), (x, 1), (x*y + 1, 1)]) +
            (1, [(y, 1), (x*y + 1, 1)]) +
            (1, [(y, 1), (x, 1), (x*y + 1, 1)])
            sage: t
            (1, [(y, 1), (x*y + 1, 1)]) + (2, [(y, 1), (x, 1), (x*y + 1, 1)])

            sage: H = x * y * (x*y + 1)
            sage: f = FFPD(1, H.factor())
            sage: g = FFPD(exp(x + y), H.factor())
            sage: s = FractionWithFactoredDenominatorSum([f, g])
            sage: s
            (1, [(y, 1), (x, 1), (x*y + 1, 1)]) +
            (e^(x + y), [(y, 1), (x, 1), (x*y + 1, 1)])
            sage: t = s._combine_like_terms_()
            sage: t
            (e^(x + y) + 1, [(y, 1), (x, 1), (x*y + 1, 1)])
        """
        if not self:
            return self

        from operator import methodcaller
        # Combine like terms.
        FFPDs = sorted(self, key=methodcaller('_total_order_key_'))
        new_FFPDs = []
        temp = FFPDs[0]
        for f in FFPDs[1:]:
            if temp.denominator_factored() == f.denominator_factored():
                # Add f to temp.
                num = temp.numerator() + f.numerator()
                temp = f.parent()(num, temp.denominator_factored())
            else:
                # Append temp to new_FFPDs and update temp.
                new_FFPDs.append(temp)
                temp = f
        new_FFPDs.append(temp)
        return FractionWithFactoredDenominatorSum(new_FFPDs)

    def sum(self):
        r"""
        Return the sum of the elements in ``self``.

        OUTPUT:

        An instance of :class:`FractionWithFactoredDenominator`.

        EXAMPLES::

            sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing, FractionWithFactoredDenominatorSum
            sage: R.<x,y> = PolynomialRing(QQ)
            sage: FFPD = FractionWithFactoredDenominatorRing(R, SR)
            sage: df = (x, 1), (y, 1), (x*y + 1, 1)
            sage: f = FFPD(2, df)
            sage: g = FFPD(2*x*y, df)
            sage: FractionWithFactoredDenominatorSum([f, g])
            (2, [(y, 1), (x, 1), (x*y + 1, 1)]) + (2, [(x*y + 1, 1)])
            sage: FractionWithFactoredDenominatorSum([f, g]).sum()
            (2, [(y, 1), (x, 1)])

            sage: f = FFPD(cos(x), [(x, 2)])
            sage: g = FFPD(cos(y), [(x, 1), (y, 2)])
            sage: FractionWithFactoredDenominatorSum([f, g])
            (cos(x), [(x, 2)]) + (cos(y), [(y, 2), (x, 1)])
            sage: FractionWithFactoredDenominatorSum([f, g]).sum()
            (y^2*cos(x) + x*cos(y), [(y, 2), (x, 2)])
        """
        if not self:
            return self

        # Compute the sum's numerator and denominator.
        R = self.denominator_ring
        summy = sum((f.quotient() for f in self))
        numer = summy.numerator()
        denom = R(summy.denominator())

        # Compute the sum's denominator factorization.
        # Could use the factor() command, but it's probably faster to use
        # the irreducible factors of the denominators of self.
        df = []  # The denominator factorization for the sum.
        if denom == 1:
            # Done
            return FractionWithFactoredDenominatorRing(numer.parent())(numer, df, reduce=False)

        factors = []
        for f in self:
            factors.extend([q for (q, e) in f.denominator_factored()])

        # Eliminate repeats from factors and sort.
        factors = sorted(list(set(factors)))

        # The irreducible factors of denom lie in factors.
        # Use this fact to build df.
        for q in factors:
            e = 0
            quo, rem = denom.quo_rem(q)
            while rem == 0:
                e += 1
                denom = quo
                quo, rem = denom.quo_rem(q)
            if e > 0:
                df.append((q, e))
        return FractionWithFactoredDenominatorRing(numer.parent())(numer, df, reduce=False)


#####################################################################
## Helper functions


def diff_prod(f_derivs, u, g, X, interval, end, uderivs, atc):
    r"""
    Take various derivatives of the equation `f = ug`,
    evaluate them at a point `c`, and solve for the derivatives of `u`.

    INPUT:

    - ``f_derivs`` -- a dictionary whose keys are all tuples of the form
      ``s + end``, where ``s`` is a sequence of variables from ``X`` whose
      length lies in ``interval``, and whose values are the derivatives
      of a function `f` evaluated at `c`
    - ``u`` -- a callable symbolic function
    - ``g`` -- an expression or callable symbolic function
    - ``X`` -- a list of symbolic variables
    - ``interval`` -- a list of positive integers
      Call the first and last values `n` and `nn`, respectively
    - ``end`` -- a possibly empty list of repetitions of the
      variable ``z``, where ``z`` is the last element of ``X``
    - ``uderivs`` -- a dictionary whose keys are the symbolic
      derivatives of order 0 to order `n-1` of ``u`` evaluated at `c`
      and whose values are the corresponding derivatives evaluated at `c`
    - ``atc`` -- a dictionary whose keys are the keys of `c` and all
      the symbolic derivatives of order 0 to order `nn` of ``g``
      evaluated `c` and whose values are the corresponding
      derivatives evaluated at `c`

    OUTPUT:

    A dictionary whose keys are the derivatives of ``u`` up to order
    `nn` and whose values are those derivatives evaluated at `c`.

    This function works by differentiating the equation `f = ug`
    with respect to the variable sequence ``s + end``,
    for all tuples ``s`` of ``X`` of lengths in ``interval``,
    evaluating at the point `c` ,
    and solving for the remaining derivatives of ``u``.
    This function assumes that ``u`` never appears in the
    differentiations of `f = ug` after evaluating at `c`.

    .. NOTE::

        For internal use by
        :meth:`FractionWithFactoredDenominator.asymptotics_multiple()`.

    EXAMPLES::

        sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import diff_prod
        sage: u = function('u')(x)
        sage: g = function('g')(x)
        sage: fd = {(x,):1,(x, x):1}
        sage: ud = {u(x=2): 1}
        sage: atc = {x: 2, g(x=2): 3, diff(g, x)(x=2): 5}
        sage: atc[diff(g, x, x)(x=2)] = 7
        sage: dd = diff_prod(fd, u, g, [x], [1, 2], [], ud, atc)
        sage: dd[diff(u, x, 2)(x=2)]
        22/9
    """
    from sage.symbolic.relation import solve

    for l in interval:
        D = {}
        rhs = []
        lhs = []
        for t in combinations_with_replacement(X, l):
            t = list(t)
            s = t + end
            lhs.append(f_derivs[tuple(s)])
            rhs.append(diff(u * g, s).subs(atc).subs(uderivs))
            # Since Sage's solve command can't take derivatives as variable
            # names, make new variables based on t to stand in for
            # diff(u, t) and store them in D.
            D[diff(u, t).subs(atc)] = var('zing' +
                                          ''.join(str(x) for x in t))
        eqns = [lhs[i] == rhs[i].subs(uderivs).subs(D)
                for i in range(len(lhs))]
        variables = D.values()
        sol = solve(eqns, *variables, solution_dict=True)
        uderivs.update(subs_all(D, sol[ZZ.zero()]))
    return uderivs


def permutation_sign(s, u):
    r"""
    This function returns the sign of the permutation on
    ``1, ..., len(u)`` that is induced by the sublist ``s`` of ``u``.

    .. NOTE::

        For internal use by
        :meth:`FractionWithFactoredDenominator.cohomology_decomposition()`.

    INPUT:

    - ``s`` -- a sublist of ``u``
    - ``u`` -- a list

    OUTPUT:

    The sign of the permutation obtained by taking indices
    within ``u`` of the list ``s + sc``, where ``sc`` is ``u``
    with the elements of ``s`` removed.

    EXAMPLES::

        sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import permutation_sign
        sage: u = ['a', 'b', 'c', 'd', 'e']
        sage: s = ['b', 'd']
        sage: permutation_sign(s, u)
        -1
        sage: s = ['d', 'b']
        sage: permutation_sign(s, u)
        1
    """
    from sage.combinat.permutation import Permutation

    # Convert lists to lists of numbers in {1,..., len(u)}
    A = [i + 1 for i in range(len(u))]
    B = [u.index(x) + 1 for x in s]

    C = sorted(set(A).difference(set(B)))
    P = Permutation(B + C)
    return P.signature()


def subs_all(f, sub, simplify=False):
    r"""
    Return the items of `f` substituted by the dictionaries
    of ``sub`` in order of their appearance in ``sub``.

    INPUT:

    - ``f`` -- an individual or list of symbolic expressions
      or dictionaries
    - ``sub`` -- an individual or list of dictionaries
    - ``simplify`` -- (default: ``False``) boolean; set to ``True`` to
      simplify the result

    OUTPUT:

    The items of ``f`` substituted by the dictionaries of ``sub`` in order
    of their appearance in ``sub``. The ``subs()`` command is used. If
    simplify is ``True``, then ``simplify()`` is used after substitution.

    EXAMPLES::

        sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import subs_all
        sage: var('x, y, z')
        (x, y, z)
        sage: a = {x:1}
        sage: b = {y:2}
        sage: c = {z:3}
        sage: subs_all(x + y + z, a)
        y + z + 1
        sage: subs_all(x + y + z, [c, a])
        y + 4
        sage: subs_all([x + y + z, y^2], b)
        [x + z + 2, 4]
        sage: subs_all([x + y + z, y^2], [b, c])
        [x + 5, 4]

    ::

        sage: var('x, y')
        (x, y)
        sage: a = {'foo': x**2 + y**2, 'bar': x - y}
        sage: b = {x: 1 , y: 2}
        sage: subs_all(a, b)
        {'bar': -1, 'foo': 5}
    """
    singleton = False
    if not isinstance(f, (list, tuple)):
        f = [f]
        singleton = True
    if not isinstance(sub, (list, tuple)):
        sub = [sub]
    g = []
    for ff in f:
        for D in sub:
            if isinstance(ff, dict):
                ff = {k: ff[k].subs(D) for k in ff}
            else:
                ff = ff.subs(D)
        g.append(ff)

    if singleton and simplify:
        if isinstance(g[0], dict):
            return g[0]
        return g[0].simplify()

    if singleton and not simplify:
        return g[0]

    if not singleton and simplify:
        G = []
        for gg in g:
            if isinstance(gg, dict):
                G.append(gg)
            else:
                G.append(gg.simplify())
        return G

    return g


def diff_all(f, V, n, ending=[], sub=None, sub_final=None,
              zero_order=0, rekey=None):
    r"""
    Return a dictionary of representative mixed partial
    derivatives of `f` from order 1 up to order `n` with respect to the
    variables in `V`.

    The default is to key the dictionary by all nondecreasing sequences
    in `V` of length 1 up to length `n`.

    INPUT:

    - ``f`` -- an individual or list of `\mathcal{C}^{n+1}` functions
    - ``V`` -- a list of variables occurring in `f`
    - ``n`` -- a natural number
    - ``ending`` -- a list of variables in `V`
    - ``sub`` -- an individual or list of dictionaries
    - ``sub_final`` -- an individual or list of dictionaries
    - ``rekey`` -- a callable symbolic function in `V` or list thereof
    - ``zero_order`` -- a natural number

    OUTPUT:

    The dictionary ``{s_1:deriv_1, ..., sr:deriv_r}``.

    Here ``s_1, ..., s_r`` is a listing of
    all nondecreasing sequences of length 1 up to length `n` over the
    alphabet `V`, where `w > v` in `X` if and only if ``str(w) > str(v)``,
    and ``deriv_j`` is the derivative of `f` with respect to the derivative
    sequence ``s_j`` and simplified with respect to the substitutions in
    ``sub`` and evaluated at ``sub_final``.
    Moreover, all derivatives with respect to sequences of length less than
    ``zero_order`` (derivatives of order less than ``zero_order`` )
    will be made zero.

    If ``rekey`` is nonempty, then ``s_1, ..., s_r`` will be replaced
    by the symbolic derivatives of the functions in ``rekey``.

    If ``ending`` is nonempty, then every derivative sequence ``s_j``
    will be suffixed by ``ending``.

    EXAMPLES::

        sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import diff_all
        sage: f = function('f')(x)
        sage: dd = diff_all(f, [x], 3)
        sage: dd[(x, x, x)]
        diff(f(x), x, x, x)

        sage: d1 = {diff(f, x): 4*x^3}
        sage: dd = diff_all(f, [x], 3, sub=d1)
        sage: dd[(x, x, x)]
        24*x

        sage: dd = diff_all(f, [x], 3, sub=d1, rekey=f)
        sage: dd[diff(f, x, 3)]
        24*x

        sage: a = {x:1}
        sage: dd = diff_all(f, [x], 3, sub=d1, rekey=f, sub_final=a)
        sage: dd[diff(f, x, 3)]
        24

    ::

        sage: X = var('x, y, z')
        sage: f = function('f')(*X)
        sage: dd = diff_all(f, X, 2, ending=[y, y, y])
        sage: dd[(z, y, y, y)]
        diff(f(x, y, z), y, y, y, z)

    ::

        sage: g = function('g')(*X)
        sage: dd = diff_all([f, g], X, 2)
        sage: dd[(0, y, z)]
        diff(f(x, y, z), y, z)

        sage: dd[(1, z, z)]
        diff(g(x, y, z), z, z)

        sage: f = exp(x*y*z)
        sage: ff = function('ff')(*X)
        sage: dd = diff_all(f, X, 2, rekey=ff)
        sage: dd[diff(ff, x, z)]
        x*y^2*z*e^(x*y*z) + y*e^(x*y*z)
    """
    singleton = False
    if not isinstance(f, list):
        f = [f]
        singleton = True

    # Build the dictionary of derivatives iteratively from a list
    # of nondecreasing derivative-order sequences.
    derivs = {}
    r = len(f)
    if ending:
        seeds = [ending]
        start = ZZ.one()
    else:
        seeds = [[v] for v in V]
        start = Integer(2)
    if singleton:
        for s in seeds:
            derivs[tuple(s)] = subs_all(diff(f[0], s), sub)
        for l in range(start, n + 1):
            for t in combinations_with_replacement(V, l):
                s = t + tuple(ending)
                derivs[s] = subs_all(diff(derivs[s[1:]], s[0]), sub)
    else:
        # Make the dictionary keys of the form (j, sequence of variables),
        # where j in range(r).
        for s in seeds:
            value = subs_all([diff(f[j], s) for j in range(r)], sub)
            derivs.update({tuple([j] + s): value[j] for j in range(r)})
        for l in range(start, n + 1):
            for t in combinations_with_replacement(V, l):
                s = t + tuple(ending)
                value = subs_all([diff(derivs[(j,) + s[1:]], s[0]) for j in range(r)], sub)
                derivs.update({(j,) + s: value[j] for j in range(r)})
    if zero_order:
        # Zero out all the derivatives of order < zero_order
        if singleton:
            for k in derivs:
                if len(k) < zero_order:
                    derivs[k] = ZZ.zero()
        else:
            # Ignore the first of element of k, which is an index.
            for k in derivs:
                if len(k) - 1 < zero_order:
                    derivs[k] = ZZ.zero()
    if sub_final:
        # Substitute sub_final into the values of derivs.
        for k in derivs:
            derivs[k] = subs_all(derivs[k], sub_final)
    if rekey:
        # Rekey the derivs dictionary by the value of rekey.
        F = rekey
        if singleton:
            # F must be a singleton.
            derivs = {diff(F, list(k)): derivs[k] for k in derivs}
        else:
            # F must be a list.
            derivs = {diff(F[k[0]], list(k)[1:]): derivs[k] for k in derivs}
    return derivs


def diff_op(A, B, AB_derivs, V, M, r, N):
    r"""
    Return the derivatives `DD^{(l+k)}(A[j] B^l)` evaluated at a point
    `p` for various natural numbers `j, k, l` which depend on `r` and `N`.

    Here `DD` is a specific second-order linear differential operator
    that depends on `M` , `A` is a list of symbolic functions,
    `B` is symbolic function, and ``AB_derivs`` contains all the derivatives
    of `A` and `B` evaluated at `p` that are necessary for the computation.

    INPUT:

    - ``A`` -- a single or length ``r`` list of symbolic functions in the
      variables ``V``
    - ``B`` -- a symbolic function in the variables ``V``.
    - ``AB_derivs`` -- a dictionary whose keys are the (symbolic)
      derivatives of ``A[0], ..., A[r-1]`` up to order ``2 * N-2`` and
      the (symbolic) derivatives of ``B`` up to order ``2 * N``;
      the values of the dictionary are complex numbers that are
      the keys evaluated at a common point `p`
    - ``V`` -- the variables of the ``A[j]`` and ``B``
    - ``M`` -- a symmetric `l \times l` matrix, where `l` is the
      length of ``V``
    - ``r, N`` -- natural numbers

    OUTPUT:

    A dictionary.

    The output is
    a dictionary whose keys are natural number tuples of the form
    `(j, k, l)`, where `l \leq 2k`, `j \leq r-1`, and `j+k \leq N-1`,
    and whose values are `DD^(l+k)(A[j] B^l)` evaluated at a point
    `p`, where `DD` is the linear second-order differential operator
    `-\sum_{i=0}^{l-1} \sum_{j=0}^{l-1} M[i][j]
    \partial^2 /(\partial V[j] \partial V[i])`.

    .. NOTE::

        For internal use by
        :meth:`FractionWithFactoredDenominator.asymptotics_smooth()` and
        :meth:`FractionWithFactoredDenominator.asymptotics_multiple()`.

    EXAMPLES::

        sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import diff_op
        sage: T = var('x, y')
        sage: A = function('A')(*tuple(T))
        sage: B = function('B')(*tuple(T))
        sage: AB_derivs = {}
        sage: M = matrix([[1, 2],[2, 1]])
        sage: DD = diff_op(A, B, AB_derivs, T, M, 1, 2)
        sage: sorted(DD)
        [(0, 0, 0), (0, 1, 0), (0, 1, 1), (0, 1, 2)]
        sage: len(DD[(0, 1, 2)])
        246
    """
    from itertools import product
    from sage.misc.mrange import xmrange

    if not isinstance(A, list):
        A = [A]

    # First, compute the necessary product derivatives of A and B.
    product_derivs = {}
    for j, k in xmrange([r, N], tuple):
        if j + k < N:
            for l in range(2 * k + 1):
                for s in combinations_with_replacement(V, 2 * (k + l)):
                    DF = diff(A[j] * B ** l, list(s)).subs(AB_derivs)
                    product_derivs[(j, k, l) + s] = DF

    # Second, compute DD^(k+l)(A[j]*B^l)(p) and store values in dictionary.
    DD = {}
    rows = M.nrows()
    for j, k in xmrange([r, N], tuple):
        if j + k < N:
            for l in range(2 * k + 1):
                # Take advantage of the symmetry of M by ignoring
                # the upper-diagonal entries of M and multiplying by
                # appropriate powers of 2.
                if k + l == 0:
                    DD[(j, k, l)] = product_derivs[(j, k, l)]
                    continue
                S = [(a, b) for a, b in xmrange([rows, rows], tuple) if b <= a]
                P = product(S, repeat=k + l)
                diffo = ZZ.zero()
                for t in P:
                    idx = (j, k, l) + diff_seq(V, t)
                    if product_derivs[idx] != ZZ.zero():
                        MM = ZZ.one()
                        for (a, b) in t:
                            MM *= M[a][b]
                            if a != b:
                                MM *= Integer(2)
                        diffo += MM * product_derivs[idx]
                DD[(j, k, l)] = (-ZZ.one()) ** (k + l) * diffo
    return DD


def diff_seq(V, s):
    r"""
    Given a list ``s`` of tuples of natural numbers, return the
    list of elements of ``V`` with indices the elements of the elements
    of ``s``.

    INPUT:

    - ``V`` -- a list
    - ``s`` -- a list of tuples of natural numbers in the interval
      ``range(len(V))``

    OUTPUT:

    The tuple ``tuple([V[tt] for tt in sorted(t)])``, where ``t`` is the
    list of elements of the elements of ``s``.

    EXAMPLES::

        sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import diff_seq
        sage: V = list(var('x, t, z'))
        sage: diff_seq(V,([0, 1],[0, 2, 1],[0, 0]))
        (x, x, x, x, t, t, z)

    .. NOTE::

        This function is for internal use by :func:`diff_op()`.
    """
    t = []
    for ss in s:
        t.extend(ss)
    return tuple([V[tt] for tt in sorted(t)])


def diff_op_simple(A, B, AB_derivs, x, v, a, N):
    r"""
    Return `DD^(e k + v l)(A B^l)` evaluated at a point `p` for
    various natural numbers `e, k, l` that depend on `v` and `N`.

    Here `DD` is a specific linear differential operator that depends
    on `a` and `v` , `A` and `B` are symbolic functions, and `AB_derivs`
    contains all the derivatives of `A` and `B` evaluated at `p` that are
    necessary for the computation.

    .. NOTE::

        For internal use by the function
        :meth:`FractionWithFactoredDenominator.asymptotics_smooth()`.

    INPUT:

    - ``A, B`` -- Symbolic functions in the variable ``x``
    - ``AB_derivs`` - a dictionary whose keys are the (symbolic)
      derivatives of ``A`` up to order ``2 * N`` if ``v`` is even or
      ``N`` if ``v`` is odd and the (symbolic) derivatives of ``B``
      up to order ``2 * N + v`` if ``v`` is even or ``N + v``
      if ``v`` is odd; the values of the dictionary are complex numbers
      that are the keys evaluated at a common point `p`
    - ``x`` -- a symbolic variable
    - ``a`` -- a complex number
    - ``v, N`` -- natural numbers

    OUTPUT:

    A dictionary.

    The output is
    a dictionary whose keys are natural number pairs of the form `(k, l)`,
    where `k < N` and `l \leq 2k` and whose values are
    `DD^(e k + v l)(A B^l)` evaluated at a point `p`.
    Here `e=2` if `v` is even, `e=1` if `v` is odd, and `DD` is the
    linear differential operator
    `(a^{-1/v} d/dt)` if `v` is even and
    `(|a|^{-1/v} i \text{sgn}(a) d/dt)` if `v` is odd.

    EXAMPLES::

        sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import diff_op_simple
        sage: A = function('A')(x)
        sage: B = function('B')(x)
        sage: AB_derivs = {}
        sage: sorted(diff_op_simple(A, B, AB_derivs, x, 3, 2, 2).items())
        [((0, 0), A(x)),
         ((1, 0), 1/2*I*2^(2/3)*diff(A(x), x)),
         ((1, 1),
          1/4*2^(2/3)*(B(x)*diff(A(x), x, x, x, x) + 4*diff(A(x), x, x, x)*diff(B(x), x) + 6*diff(A(x), x, x)*diff(B(x), x, x) + 4*diff(A(x), x)*diff(B(x), x, x, x) + A(x)*diff(B(x), x, x, x, x)))]
    """
    from sage.functions.other import sqrt

    I = sqrt(-ZZ.one())
    DD = {}
    if v.mod(Integer(2)) == ZZ.zero():
        for k in range(N):
            for l in range(2 * k + 1):
               DD[(k, l)] = ((a ** (-ZZ.one() / v)) ** (2 * k + v * l) *
                              diff(A * B ** l, x,
                                   2 * k + v * l).subs(AB_derivs))
    else:
        for k in range(N):
            for l in range(k + 1):
                DD[(k, l)] = ((abs(a) ** (-ZZ.one() / v) * I *
                               a / abs(a)) ** (k + v * l) *
                              diff(A * B ** l, x,
                                   k + v * l).subs(AB_derivs))
    return DD


def direction(v, coordinate=None):
    r"""
    Return ``[vv/v[coordinate] for vv in v]`` where
    ``coordinate`` is the last index of ``v`` if not specified otherwise.

    INPUT:

    - ``v`` -- a vector
    - ``coordinate`` -- (optional; default: ``None``) an index for ``v``

    EXAMPLES::

        sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import direction
        sage: direction([2, 3, 5])
        (2/5, 3/5, 1)
        sage: direction([2, 3, 5], 0)
        (1, 3/2, 5/2)
    """
    if coordinate is None:
        coordinate = len(v) - 1
    return tuple([vv / v[coordinate] for vv in v])


def coerce_point(R, p):
    r"""
    Coerce the keys of the dictionary ``p`` into the ring ``R``.

    .. WARNING::

        This method assumes that it is possible.

    EXAMPLES::

        sage: from sage.rings.asymptotic.asymptotics_multivariate_generating_functions import FractionWithFactoredDenominatorRing, coerce_point
        sage: R.<x,y> = PolynomialRing(QQ)
        sage: FFPD = FractionWithFactoredDenominatorRing(R)
        sage: f = FFPD()
        sage: p = {SR(x): 1, SR(y): 7/8}
        sage: for k in sorted(p, key=str):
        ....:     print("{} {} {}".format(k, k.parent(), p[k]))
        x Symbolic Ring 1
        y Symbolic Ring 7/8
        sage: q = coerce_point(R, p)
        sage: for k in sorted(q, key=str):
        ....:     print("{} {} {}".format(k, k.parent(), q[k]))
        x Multivariate Polynomial Ring in x, y over Rational Field 1
        y Multivariate Polynomial Ring in x, y over Rational Field 7/8
    """
    if p and list(p)[0].parent() != R:
        try:
            return {x: p[SR(x)] for x in R.gens()}
        except TypeError:
            pass
    return p
