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](..).

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$$\mathrm{with}\left(\mathrm{ChainTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[z\,y\,x\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{pz}≔x{z}^2+y^2+1$

$\mathrm{pz}≔x{z}^2+y^2+1$

$\mathrm{px}≔x^2+1$

$\mathrm{px}≔x^2+1$

$\mathrm{rc}≔\mathrm{Chain}\left(\left[\mathrm{px}\,\mathrm{pz}\right]\,\mathrm{Empty}\left(R\right)\,R\right)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{Lp}≔\left[\mathrm{pz}\,\mathrm{px}\,\mathrm{pz}+\mathrm{px}\,x^2\mathrm{pz}-\mathrm{px}\,{\mathrm{px}}^2\,3\mathrm{px}+xy\mathrm{pz}\right]$

$\mathrm{Lp}≔\left[z^{2}x+y^2+1\,x^2+1\,z^{2}x+x^2+y^2+2\,\left(z^{2}x+y^2+1\right)x^2-x^2-1\,{\left(x^2+1\right)}^2\,\left(z^{2}x+y^2+1\right)xy+3x^2+3\right]$

$$\begin{gathered} \mathbf{for}i\mathbf{to}\mathrm{nops}\left(\mathrm{Lp}\right)\mathbf{do} \\ \mathrm{IsInSaturate}\left(\mathrm{Lp}\left[i\right]\,\mathrm{rc}\,R\right) \\ \mathbf{end}\mathbf{do} \end{gathered}$$

$\mathrm{true}$

$\mathrm{true}$

$\mathrm{true}$

$\mathrm{true}$

$\mathrm{true}$

$\mathrm{true}$