Sylvester matrix of two polynomials : sylvester

5.30.18 Sylvester matrix of two polynomials : sylvester

sylvester takes two polynomials as arguments.
sylvester returns the Sylvester matrix S of these polynomials.
If A(x)=∑_i=0ⁱ⁼ⁿ a_ixⁱ and B(x)=∑_i=0^i=mb_ixⁱ are 2 polynomials, their Sylvester matrix S is a square matrix of size m+n where m=degree(B(x)) and n=degree(A(x)). The m first lines are made with the A(x) coefficients, so that :

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s₁₁=a_n	s₁₂=a_n−1	⋯	s_1(n+1)=a₀	0	⋯	0
s₂₁=0	s₂₂=a_n	⋯	s_2(n+1)=a₁	s_2(n+2)=a₀	⋯	0
⋮	⋮	⋮	⋱	⋮	⋱	⋮
s_m1=0	s_m2=0	⋯	s_m(n+1)=a_m−1	s_m(n+2)=a_m−2	⋯	a₀

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and the n further lines are made with the B(x) coefficients, so that :

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s_(m+1)1=b_m	s_(m+1)2=b_m−1	⋯	s_(m+1)(m+1)=b₀	0	⋯	0
⋮	⋮	⋮	⋱	⋮	⋱	⋮
s_(m+n)1=0	s_(m+n)2=0	⋯	s_(m+n)(m+1)=b_n−1	b_n−2	⋯	b₀

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Input :

sylvester(x^3-p*x+q,3*x^2-p,x)

Output :

[[1,0,-p,q,0],[0,1,0,-p,q],[3,0,-p,0,0], [0,3,0,-p,0],[0,0,3,0,-p]]

Input :

det([[1,0,-p,q,0],[0,1,0,-p,q],[3,0,-p,0,0], [0,3,0,-p,0],[0,0,3,0,-p]])

Output :

-4*p^3--27*q^2