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 decreassing order). If the polynomials are expressions, an optionnal 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 :

$$\tt\left\{ \begin{array}{rlr} P(x)=&x\ &\bmod\ (x^2+1)\\ P(x)=&x-1\ &\bmod\ (x^2-1) \end{array}\right.$$

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) = - $${\frac{{x^2-2.x+1}}{{2}}}$$ ( 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]