The command IsInRadical(p, rc, R) returns true if and only if p belongs to the radical of the saturated ideal of rc.

This command is part of the RegularChains[ChainTools] package, so it can be used in the form IsInRadical(..) 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][IsInRadical](..).

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

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

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

$R≔\mathrm{polynomial\_ring}$

$\mathrm{sys}≔\left\{x^2+1\,{\left(y+2x\right)}^2\right\}$

$\mathrm{sys}≔\left\{{\left(y+2x\right)}^2\,x^2+1\right\}$

$\mathrm{out}≔\mathrm{Triangularize}\left(\mathrm{sys}\,R\right)\;$$\mathrm{rc}≔\mathrm{out}\left[1\right]$

$\mathrm{out}≔\left[\mathrm{regular\_chain}\right]$

$\mathrm{rc}≔\mathrm{regular\_chain}$

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

$\left[y^2+4xy-4\,x^2+1\right]$

$\mathrm{NumberOfSolutions}\left(\mathrm{rc}\,R\right)$

$4$

$\mathrm{IsInSaturate}\left(y+2x\,\mathrm{rc}\,R\right)$

$\mathrm{false}$

$\mathrm{IsInRadical}\left(y+2x\,\mathrm{rc}\,R\right)$

$\mathrm{true}$

$\mathrm{out}≔\mathrm{Triangularize}\left(\mathrm{sys}\,R\,\mathrm{radical}=\mathrm{yes}\right)\;$$\mathrm{sfrc}≔\mathrm{out}\left[1\right]$

$\mathrm{out}≔\left[\mathrm{regular\_chain}\right]$

$\mathrm{sfrc}≔\mathrm{regular\_chain}$

$\mathrm{Equations}\left(\mathrm{sfrc}\,R\right)\;$$\mathrm{NumberOfSolutions}\left(\mathrm{sfrc}\,R\right)$

$\left[y+2x\,x^2+1\right]$

$2$

$\mathrm{IsInSaturate}\left(y+2x\,\mathrm{sfrc}\,R\right)$

$\mathrm{true}$