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.
We consider two bivariate polynomials and want to compute their common solutions
We first compute their subresultant chain using FFT techniques
We deduce their resultants
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
We compute the GCD of f1 and f2 modulo r2
We normalize this GCD w.r.t. r2 which leads to a simpler expression with one as leading coefficient
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