IsInSaturate - Maple Help

RegularChains[ChainTools]

 IsInSaturate
 test membership to the saturated ideal of a regular chain

 Calling Sequence IsInSaturate(p, rc, R)

Parameters

 p - polynomial of R rc - regular chain of R R - polynomial ring

Description

 • The command IsInSaturate(p,rc,R) returns true if and only if p belongs to the saturated ideal of rc.
 • This command is part of the RegularChains[ChainTools] package, so it can be used in the form IsInSaturate(..) only after executing the command with(RegularChains[ChainTools]).  However, it can always be accessed through the long form of the command by using RegularChains[ChainTools][IsInSaturate](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$$\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[z,y,x\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $\mathrm{pz}≔x{z}^{2}+{y}^{2}+1$
 ${\mathrm{pz}}{≔}{x}{}{{z}}^{{2}}{+}{{y}}^{{2}}{+}{1}$ (2)
 > $\mathrm{px}≔{x}^{2}+1$
 ${\mathrm{px}}{≔}{{x}}^{{2}}{+}{1}$ (3)
 > $\mathrm{rc}≔\mathrm{Chain}\left(\left[\mathrm{px},\mathrm{pz}\right],\mathrm{Empty}\left(R\right),R\right)$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (4)
 > $\mathrm{Lp}≔\left[\mathrm{pz},\mathrm{px},\mathrm{pz}+\mathrm{px},{x}^{2}\mathrm{pz}-\mathrm{px},{\mathrm{px}}^{2},3\mathrm{px}+xy\mathrm{pz}\right]$
 ${\mathrm{Lp}}{≔}\left[{{z}}^{{2}}{}{x}{+}{{y}}^{{2}}{+}{1}{,}{{x}}^{{2}}{+}{1}{,}{{z}}^{{2}}{}{x}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{2}{,}\left({{z}}^{{2}}{}{x}{+}{{y}}^{{2}}{+}{1}\right){}{{x}}^{{2}}{-}{{x}}^{{2}}{-}{1}{,}{\left({{x}}^{{2}}{+}{1}\right)}^{{2}}{,}\left({{z}}^{{2}}{}{x}{+}{{y}}^{{2}}{+}{1}\right){}{x}{}{y}{+}{3}{}{{x}}^{{2}}{+}{3}\right]$ (5)
 > $\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}i\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{to}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{nops}\left(\mathrm{Lp}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{IsInSaturate}\left(\mathrm{Lp}\left[i\right],\mathrm{rc},R\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}$
 ${\mathrm{true}}$
 ${\mathrm{true}}$
 ${\mathrm{true}}$
 ${\mathrm{true}}$
 ${\mathrm{true}}$
 ${\mathrm{true}}$ (6)