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5.32.4  Build a polynomial from its evaluation: genpoly

indexgenpoly genpoly takes three arguments: a polynomial P with n−1 variables, an integer b and the name of a variable var.
genpoly returns the polynomial Q with n variables (the P variables and the variable var given as second argument), such that:

In other words, P is written in base b but using the convention that the euclidean remainder belongs to ]−b/2 ; b/2] (this convention is also known as s-mod representation). Input:

genpoly(61,6,x)

Output:

2*x^2-2*x+1

Indeed 61 divided by 6 is 10 with remainder 1, then 10 divided by 6 is 2 with remainder -2 (instead of the usual quotient 1 and remainder 4 out of bounds),

61=2*62−2*6+1 

Input:

genpoly(5,6,x)

Output:

x-1

Indeed: 5=6−1
Input:

genpoly(7,6,x)

Output:

x+1

Indeed: 7=6+1
Input:

genpoly(7*y+5,6,x)

Output:

x*y+x+y-1

Indeed: x*y+x+y−1=y(x+1)+(x−1)
Input:

genpoly(7*y+5*z^2,6,x)

Output:

x*y+x*z+y-z

Indeed: x*y+x*z+yz=y*(x+1)+z*(x−1)


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