The command IsRegular(in_p, in_rc, R) returns true if and only if p is regular modulo rc, that is if and only if p is not a zero-divisor modulo the saturated ideal of rc.

This command is part of the RegularChains package, so it can be used in the form IsRegular(..) only after executing the command with(RegularChains).  However, it can always be accessed through the long form of the command by using RegularChains[IsRegular](..).

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

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

$R≔\mathrm{polynomial\_ring}$

$T≔\mathrm{Empty}\left(R\right)$

$T≔\mathrm{regular\_chain}$

$T≔\mathrm{Chain}\left(\left[\left(z+1\right)\left(z+2\right)\,y^2+z\,\left(x-z\right)\left(x-y\right)\right]\,T\,R\right)$

$T≔\mathrm{regular\_chain}$

$\mathrm{Equations}\left(T\,R\right)$

$\left[x^2+\left(-y-z\right)x+zy\,y^2+z\,z^2+3z+2\right]$

$p≔\left(z+1\right)\left(x^3+5\right)$

$p≔\left(z+1\right)\left(x^3+5\right)$

$\mathrm{IsRegular}\left(p\,T\,R\right)$

$\mathrm{false}$

$\mathrm{regl},\mathrm{zdl}≔\mathrm{op}\left(\mathrm{Regularize}\left(p\,T\,R\right)\right)$

$\mathrm{regl},\mathrm{zdl}≔\left[\mathrm{regular\_chain}\right],\left[\mathrm{regular\_chain}\,\mathrm{regular\_chain}\,\mathrm{regular\_chain}\right]$

$\mathrm{map}\left(\mathrm{Equations}\,\mathrm{zdl}\,R\right)$

$\left[\left[x-1\,y-1\,z+1\right]\,\left[x+1\,y-1\,z+1\right]\,\left[x+1\,y+1\,z+1\right]\right]$

$q≔x+y+z$

$q≔x+y+z$

$\mathrm{IsRegular}\left(q\,T\,R\right)$

$\mathrm{true}$

$\mathrm{Regularize}\left(q\,T\,R\right)$

$\left[\left[T\right]\,\left[\right]\right]$

$\mathrm{Inverse}\left(q\,T\,R\right)$

$\left[\left[\left[-235xy{z}^2+94x{z}^3-515xyz+112x{z}^2-206zx\,504\,\mathrm{regular\_chain}\right]\,\left[-235xy{z}^2+94x{z}^3-515xyz+112x{z}^2-206zx\,504\,\mathrm{regular\_chain}\right]\,\left[-235xy{z}^2+94x{z}^3+470y^2{z}^2+282y{z}^3-188z^4-515xyz+112x{z}^2+1030y^{2}z+806y{z}^2-224z^3-206zx+412zy+412z^2\,504\,\mathrm{regular\_chain}\right]\,\left[-235xy{z}^2+94x{z}^3+470y^2{z}^2+282y{z}^3-188z^4-515xyz+112x{z}^2+1030y^{2}z+806y{z}^2-224z^3-206zx+412zy+412z^2\,504\,\mathrm{regular\_chain}\right]\right]\,\left[\right]\right]$

$r≔z-x$

$r≔-x+z$

$\mathrm{IsRegular}\left(r\,T\,R\right)$

$\mathrm{false}$

$\mathrm{Regularize}\left(r\,T\,R\right)$

$\left[\left[\mathrm{regular\_chain}\,\mathrm{regular\_chain}\right]\,\left[\mathrm{regular\_chain}\,\mathrm{regular\_chain}\right]\right]$

$\mathrm{Inverse}\left(r\,T\,R\right)$

$\left[\left[\left[\mathrm{-1}\,2\,\mathrm{regular\_chain}\right]\,\left[y+z\,2\,\mathrm{regular\_chain}\right]\right]\,\left[\mathrm{regular\_chain}\,\mathrm{regular\_chain}\right]\right]$