normalize a polynomial w.r.t a 0-dim regular chain
NormalizePolynomialDim0(f, rc, R)
a polynomial ring
a regular chain of R
polynomial of R
The command NormalizePolynomialDim0 returns a normalized form of f w.r.t. rc, that is, a polynomial q which is associated to f modulo rc, such that q is normalized w.r.t. rc.
rc is zero-dimensional regular chain, and f together with rc forms a zero-dimensional regular chain.
Moreover R must have a prime characteristic p such that FFT-based polynomial arithmetic can be used for this actual computation. The higher the degrees of f and rc are, the larger must be e such that 2e divides p−1. If the degree of f or rc is too large, then an error is raised.
p ≔ 962592769:
vars ≔ y,x:
R ≔ PolynomialRing⁡vars,p:
We consider two bivariate polynomials and want to compute their common solutions
f1 ≔ x⁢y2+y+1+2:
f2 ≔ x+1⁢y2+y+1+x3+x+1:
We first compute their subresultant chain using FFT techniques
SCube ≔ SubresultantChainSpecializationCube⁡f1,f2,y,R,1
We deduce their resultants
r2 ≔ ResultantBySpecializationCube⁡f1,f2,x,SCube,R
We observe below that no root of r2 cancels the leading coefficients of f1 or f2. Hence, any roots of r2 can be extended into a solution of the system by a GCD computation.
We define the regular chain consisting of r2
rc ≔ Chain⁡r2,Empty⁡R,R
We compute the GCD of f1 and f2 modulo r2
g2 ≔ RegularGcdBySpecializationCube⁡f1,f2,rc,SCube,R
We normalize this GCD w.r.t. r2 which leads to a simpler expression with one as leading coefficient
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