NormalFormDim0 - Maple Help

RegularChains[FastArithmeticTools]

 NormalFormDim0
 normal form of a polynomial w.r.t. a 0-dim regular chain

 Calling Sequence NormalFormDim0(f, rc, R)

Parameters

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

Description

 • Returns the  normal form of f w.r.t. rc in the sense of Groebner bases
 • rc must be a normalized zero-dimensional regular chain and all variables in f must be algebraic w.r.t. rc. See the subpackage ChainTools for these notions.
 • 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.
 • The algorithm relies on the fast division trick (based on power series inversion) and FFT-based multivariate multiplication. When both commands NormalFormDim0 and NormalForm apply, the former one will often outperform the latter one.

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $\mathrm{with}\left(\mathrm{FastArithmeticTools}\right):$
 > $p≔962592769:$
 > $\mathrm{vars}≔\left[x,y,z\right]:$
 > $N≔\mathrm{nops}\left(\mathrm{vars}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\mathrm{vars},p\right):$
 > $\mathrm{c1}≔{z}^{2}+286748183z+134705551:$
 > $\mathrm{c2}≔{y}^{2}+914706789yz+686506773y+823875308z+417988453:$
 > $\mathrm{c3}≔{x}^{2}+224047618xyz+197329999xy+838274835xz+792563861x+600038529yz+434770098y+251400283z+918968185:$
 > $\mathrm{lf}≔\left[\mathrm{c1},\mathrm{c2},\mathrm{c3}\right]:$
 > $\mathrm{tc}≔\mathrm{Chain}\left(\mathrm{lf},\mathrm{Empty}\left(R\right),R\right):$
 > $\mathrm{lf}≔\left[x+\mathrm{c1}\mathrm{c2}\mathrm{c3},\left(\mathrm{c1}-\mathrm{c2}\right)\left(\mathrm{c1}-\mathrm{c3}\right),1+\mathrm{c1}\left(\mathrm{c2}-\mathrm{c3}\right)\right]:$
 > $\mathrm{dg}≔10:$
 > $\mathrm{q1}≔\mathrm{randpoly}\left(\mathrm{vars},\mathrm{dense},\mathrm{degree}=\mathrm{dg}\right)+\left(\mathrm{rand}\left(\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p:$
 > $\mathrm{q2}≔\mathrm{randpoly}\left(\mathrm{vars},\mathrm{dense},\mathrm{degree}=\mathrm{dg}\right)+\left(\mathrm{rand}\left(\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p:$
 > $\mathrm{q3}≔\mathrm{randpoly}\left(\mathrm{vars},\mathrm{dense},\mathrm{degree}=\mathrm{dg}\right)+\left(\mathrm{rand}\left(\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p:$
 > $r≔\mathrm{randpoly}\left(\mathrm{vars},\mathrm{dense},\mathrm{degree}=\mathrm{dg}\right)+\left(\mathrm{rand}\left(\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p:$
 > $f≔\mathrm{q1}\mathrm{c1}+\mathrm{q2}\mathrm{c2}+\mathrm{q3}\mathrm{c3}+r$
 ${f}{≔}{610303894}{+}\left({{z}}^{{2}}{+}{286748183}{}{z}{+}{134705551}\right){}\left({962592762}{}{{x}}^{{10}}{+}{22}{}{{x}}^{{9}}{}{y}{+}{962592714}{}{{x}}^{{9}}{}{z}{+}{87}{}{{x}}^{{8}}{}{{y}}^{{2}}{+}{962592713}{}{{x}}^{{8}}{}{y}{}{z}{+}{962592707}{}{{x}}^{{8}}{}{{z}}^{{2}}{+}{962592765}{}{{x}}^{{7}}{}{{y}}^{{3}}{+}{962592686}{}{{x}}^{{7}}{}{{y}}^{{2}}{}{z}{+}{62}{}{{x}}^{{7}}{}{y}{}{{z}}^{{2}}{+}{962592725}{}{{x}}^{{7}}{}{{z}}^{{3}}{+}{962592759}{}{{x}}^{{6}}{}{{y}}^{{4}}{+}{962592762}{}{{x}}^{{6}}{}{{y}}^{{3}}{}{z}{+}{42}{}{{x}}^{{6}}{}{{y}}^{{2}}{}{{z}}^{{2}}{+}{75}{}{{x}}^{{6}}{}{y}{}{{z}}^{{3}}{+}{72}{}{{x}}^{{6}}{}{{z}}^{{4}}{+}{29}{}{{x}}^{{5}}{}{{y}}^{{5}}{+}{98}{}{{x}}^{{5}}{}{{y}}^{{4}}{}{z}{+}{10}{}{{x}}^{{5}}{}{{y}}^{{3}}{}{{z}}^{{2}}{+}{962592740}{}{{x}}^{{5}}{}{{y}}^{{2}}{}{{z}}^{{3}}{+}{962592722}{}{{x}}^{{5}}{}{y}{}{{z}}^{{4}}{+}{962592759}{}{{x}}^{{5}}{}{{z}}^{{5}}{+}{{x}}^{{4}}{}{{y}}^{{6}}{+}{55}{}{{x}}^{{4}}{}{{y}}^{{5}}{}{z}{+}{16}{}{{x}}^{{4}}{}{{y}}^{{4}}{}{{z}}^{{2}}{+}{962592754}{}{{x}}^{{4}}{}{{y}}^{{3}}{}{{z}}^{{3}}{+}{962592682}{}{{x}}^{{4}}{}{{y}}^{{2}}{}{{z}}^{{4}}{+}{962592678}{}{{x}}^{{4}}{}{y}{}{{z}}^{{5}}{+}{13}{}{{x}}^{{4}}{}{{z}}^{{6}}{+}{962592709}{}{{x}}^{{3}}{}{{y}}^{{7}}{+}{962592686}{}{{x}}^{{3}}{}{{y}}^{{6}}{}{z}{+}{962592721}{}{{x}}^{{3}}{}{{y}}^{{5}}{}{{z}}^{{2}}{+}{37}{}{{x}}^{{3}}{}{{y}}^{{4}}{}{{z}}^{{3}}{+}{25}{}{{x}}^{{3}}{}{{y}}^{{3}}{}{{z}}^{{4}}{+}{64}{}{{x}}^{{3}}{}{{y}}^{{2}}{}{{z}}^{{5}}{+}{962592767}{}{{x}}^{{3}}{}{y}{}{{z}}^{{6}}{+}{33}{}{{x}}^{{3}}{}{{z}}^{{7}}{+}{50}{}{{x}}^{{2}}{}{{y}}^{{8}}{+}{962592709}{}{{x}}^{{2}}{}{{y}}^{{7}}{}{z}{+}{7}{}{{x}}^{{2}}{}{{y}}^{{6}}{}{{z}}^{{2}}{+}{34}{}{{x}}^{{2}}{}{{y}}^{{5}}{}{{z}}^{{3}}{+}{52}{}{{x}}^{{2}}{}{{y}}^{{4}}{}{{z}}^{{4}}{+}{69}{}{{x}}^{{2}}{}{{y}}^{{3}}{}{{z}}^{{5}}{+}{962592734}{}{{x}}^{{2}}{}{{y}}^{{2}}{}{{z}}^{{6}}{+}{962592700}{}{{x}}^{{2}}{}{y}{}{{z}}^{{7}}{+}{962592759}{}{{x}}^{{2}}{}{{z}}^{{8}}{+}{962592747}{}{x}{}{{y}}^{{9}}{+}{51}{}{x}{}{{y}}^{{8}}{}{z}{+}{50}{}{x}{}{{y}}^{{7}}{}{{z}}^{{2}}{+}{962592722}{}{x}{}{{y}}^{{6}}{}{{z}}^{{3}}{+}{25}{}{x}{}{{y}}^{{5}}{}{{z}}^{{4}}{+}{10}{}{x}{}{{y}}^{{4}}{}{{z}}^{{5}}{+}{31}{}{x}{}{{y}}^{{3}}{}{{z}}^{{6}}{+}{962592718}{}{x}{}{{y}}^{{2}}{}{{z}}^{{7}}{+}{962592733}{}{x}{}{y}{}{{z}}^{{8}}{+}{81}{}{x}{}{{z}}^{{9}}{+}{962592766}{}{{y}}^{{10}}{+}{962592713}{}{{y}}^{{9}}{}{z}{+}{962592699}{}{{y}}^{{8}}{}{{z}}^{{2}}{+}{962592748}{}{{y}}^{{7}}{}{{z}}^{{3}}{+}{962592718}{}{{y}}^{{6}}{}{{z}}^{{4}}{+}{45}{}{{y}}^{{5}}{}{{z}}^{{5}}{+}{86}{}{{y}}^{{4}}{}{{z}}^{{6}}{+}{62}{}{{y}}^{{3}}{}{{z}}^{{7}}{+}{61}{}{{y}}^{{2}}{}{{z}}^{{8}}{+}{962592698}{}{y}{}{{z}}^{{9}}{+}{962592709}{}{{z}}^{{10}}{+}{962592675}{}{{x}}^{{9}}{+}{97}{}{{x}}^{{8}}{}{z}{+}{962592759}{}{{x}}^{{7}}{}{{y}}^{{2}}{+}{962592687}{}{{x}}^{{7}}{}{y}{}{z}{+}{71}{}{{x}}^{{7}}{}{{z}}^{{2}}{+}{962592729}{}{{x}}^{{6}}{}{{y}}^{{3}}{+}{962592719}{}{{x}}^{{6}}{}{{y}}^{{2}}{}{z}{+}{962592677}{}{{x}}^{{6}}{}{y}{}{{z}}^{{2}}{+}{37}{}{{x}}^{{6}}{}{{z}}^{{3}}{+}{962592746}{}{{x}}^{{5}}{}{{y}}^{{4}}{+}{962592708}{}{{x}}^{{5}}{}{{y}}^{{3}}{}{z}{+}{95}{}{{x}}^{{5}}{}{{y}}^{{2}}{}{{z}}^{{2}}{+}{40}{}{{x}}^{{5}}{}{y}{}{{z}}^{{3}}{+}{31}{}{{x}}^{{5}}{}{{z}}^{{4}}{+}{962592741}{}{{x}}^{{4}}{}{{y}}^{{5}}{+}{30}{}{{x}}^{{4}}{}{{y}}^{{4}}{}{z}{+}{962592710}{}{{x}}^{{4}}{}{{y}}^{{3}}{}{{z}}^{{2}}{+}{47}{}{{x}}^{{4}}{}{{y}}^{{2}}{}{{z}}^{{3}}{+}{962592681}{}{{x}}^{{4}}{}{y}{}{{z}}^{{4}}{+}{962592759}{}{{x}}^{{4}}{}{{z}}^{{5}}{+}{98}{}{{x}}^{{3}}{}{{y}}^{{6}}{+}{962592750}{}{{x}}^{{3}}{}{{y}}^{{5}}{}{z}{+}{5}{}{{x}}^{{3}}{}{{y}}^{{4}}{}{{z}}^{{2}}{+}{91}{}{{x}}^{{3}}{}{{y}}^{{3}}{}{{z}}^{{3}}{+}{962592679}{}{{x}}^{{3}}{}{{y}}^{{2}}{}{{z}}^{{4}}{+}{71}{}{{x}}^{{3}}{}{y}{}{{z}}^{{5}}{+}{10}{}{{x}}^{{3}}{}{{z}}^{{6}}{+}{962592727}{}{{x}}^{{2}}{}{{y}}^{{7}}{+}{962592680}{}{{x}}^{{2}}{}{{y}}^{{6}}{}{z}{+}{962592701}{}{{x}}^{{2}}{}{{y}}^{{5}}{}{{z}}^{{2}}{+}{962592749}{}{{x}}^{{2}}{}{{y}}^{{4}}{}{{z}}^{{3}}{+}{80}{}{{x}}^{{2}}{}{{y}}^{{3}}{}{{z}}^{{4}}{+}{97}{}{{x}}^{{2}}{}{{y}}^{{2}}{}{{z}}^{{5}}{+}{962592723}{}{{x}}^{{2}}{}{y}{}{{z}}^{{6}}{+}{962592704}{}{{x}}^{{2}}{}{{z}}^{{7}}{+}{962592742}{}{x}{}{{y}}^{{8}}{+}{60}{}{x}{}{{y}}^{{7}}{}{z}{+}{962592672}{}{x}{}{{y}}^{{6}}{}{{z}}^{{2}}{+}{31}{}{x}{}{{y}}^{{5}}{}{{z}}^{{3}}{+}{962592763}{}{x}{}{{y}}^{{4}}{}{{z}}^{{4}}{+}{73}{}{x}{}{{y}}^{{3}}{}{{z}}^{{5}}{+}{88}{}{x}{}{{y}}^{{2}}{}{{z}}^{{6}}{+}{962592712}{}{x}{}{y}{}{{z}}^{{7}}{+}{65}{}{x}{}{{z}}^{{8}}{+}{962592678}{}{{y}}^{{9}}{+}{42}{}{{y}}^{{8}}{}{z}{+}{962592742}{}{{y}}^{{7}}{}{{z}}^{{2}}{+}{16}{}{{y}}^{{6}}{}{{z}}^{{3}}{+}{49}{}{{y}}^{{5}}{}{{z}}^{{4}}{+}{962592672}{}{{y}}^{{4}}{}{{z}}^{{5}}{+}{96}{}{{y}}^{{3}}{}{{z}}^{{6}}{+}{962592767}{}{{y}}^{{2}}{}{{z}}^{{7}}{+}{962592703}{}{y}{}{{z}}^{{8}}{+}{7}{}{{z}}^{{9}}{+}{962592696}{}{{x}}^{{8}}{+}{80}{}{{x}}^{{7}}{}{y}{+}{962592752}{}{{x}}^{{7}}{}{z}{+}{23}{}{{x}}^{{6}}{}{{y}}^{{2}}{+}{6}{}{{x}}^{{6}}{}{y}{}{z}{+}{962592746}{}{{x}}^{{6}}{}{{z}}^{{2}}{+}{962592761}{}{{x}}^{{5}}{}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