NormalizePolynomialDim0 - Maple Help

RegularChains[FastArithmeticTools]

 NormalizePolynomialDim0
 normalize a polynomial w.r.t a 0-dim regular chain

 Calling Sequence NormalizePolynomialDim0(f, rc, R)

Parameters

 R - a polynomial ring rc - a regular chain of R f - polynomial of R

Description

 • 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 ${2}^{e}$ divides $p-1$.  If the degree of f or rc is too large, then an error is raised.

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{FastArithmeticTools}\right):$
 > $\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $p≔962592769:$
 > $\mathrm{vars}≔\left[y,x\right]:$
 > $R≔\mathrm{PolynomialRing}\left(\mathrm{vars},p\right):$

We consider two bivariate polynomials and want to compute their common solutions

 > $\mathrm{f1}≔x\left({y}^{2}+y+1\right)+2:$
 > $\mathrm{f2}≔\left(x+1\right)\left({y}^{2}+y+1\right)+{x}^{3}+x+1:$

We first compute their subresultant chain using FFT techniques

 > $\mathrm{SCube}≔\mathrm{SubresultantChainSpecializationCube}\left(\mathrm{f1},\mathrm{f2},y,R,1\right)$
 ${\mathrm{SCube}}{≔}{\mathrm{subresultant_chain_specialization_cube}}$ (1)

We deduce their resultants

 > $\mathrm{r2}≔\mathrm{ResultantBySpecializationCube}\left(\mathrm{f1},\mathrm{f2},x,\mathrm{SCube},R\right)$
 ${\mathrm{r2}}{≔}{{x}}^{{8}}{+}{2}{}{{x}}^{{6}}{+}{962592767}{}{{x}}^{{5}}{+}{962592766}{}{{x}}^{{4}}{+}{962592767}{}{{x}}^{{3}}{+}{962592766}{}{{x}}^{{2}}{+}{4}{}{x}{+}{4}$ (2)

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.

 > $\mathrm{Gcd}\left(\mathrm{r2},x\left(x+1\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p$
 ${1}$ (3)

We define the regular chain consisting of r2

 > $\mathrm{rc}≔\mathrm{Chain}\left(\left[\mathrm{r2}\right],\mathrm{Empty}\left(R\right),R\right)$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (4)

We compute the GCD of f1 and f2 modulo r2

 > $\mathrm{g2}≔\mathrm{RegularGcdBySpecializationCube}\left(\mathrm{f1},\mathrm{f2},\mathrm{rc},\mathrm{SCube},R\right)$
 ${\mathrm{g2}}{≔}\left[\left[{{x}}^{{3}}{+}{x}{}{{y}}^{{2}}{+}{x}{}{y}{+}{{y}}^{{2}}{+}{2}{}{x}{+}{y}{+}{2}{,}{\mathrm{regular_chain}}\right]{,}\left[{{x}}^{{3}}{+}{x}{}{{y}}^{{2}}{+}{x}{}{y}{+}{{y}}^{{2}}{+}{2}{}{x}{+}{y}{+}{2}{,}{\mathrm{regular_chain}}\right]\right]$ (5)

We normalize this GCD w.r.t. r2 which leads to a simpler expression with one as leading coefficient

 > $\mathrm{NormalizePolynomialDim0}\left(\mathrm{g2}\left[1\right]\left[1\right],\mathrm{g2}\left[1\right]\left[2\right],R\right)$
 ${{x}}^{{3}}{+}{{y}}^{{2}}{+}{x}{+}{y}$ (6)