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5.30.13  Chinese remainders: chinrem

chinrem takes two lists as argument, each list being made of 2 polynomials (either expressions or as a list of coefficients in decreasing order). If the polynomials are expressions, an optional third argument may be provided to specify the main variable, by default x is used. chinrem([A,R],[B,Q]) returns the list of two polynomials P and S such that:

S=R Q,     P=A (mod R ),    P=B (mod Q ) 

If R and Q are coprime, a solution P always exists and all the solutions are congruent modulo S=R*Q. For example, assume we want to solve:



P(x)=x mod (x2+1)
      P(x)=x−1 mod (x2−1) 
 

Input:

chinrem([[1,0],[1,0,1]],[[1,-1],[1,0,-1]])

Output:

[[1/-2,1,1/-2],[1,0,0,0,-1]]

or:

chinrem([x,x^2+1],[x-1,x^2-1])

Output:

[1/-2*x^2+x+1/-2,x^4-1]

hence P(x)=−x2−2.x+1/2 (mod x4−1)
Another example, input:

chinrem([[1,2],[1,0,1]],[[1,1],[1,1,1]])

Output:

[[-1,-1,0,1],[1,1,2,1,1]]

or:

chinrem([y+2,y^2+1],[y+1,y^2+y+1],y)

Output:

[-y^3-y^2+1,y^4+y^3+2*y^2+y+1]

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