The command JacobianMatrix(sys, rc, R) returns the Jacobian matrix of sys modulo the saturated ideal of rc.

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

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

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

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

$R≔\mathrm{polynomial\_ring}$

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

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

$\mathrm{dec}≔\mathrm{Triangularize}\left(\mathrm{sys}\,R\right)$

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

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

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

$\mathrm{seq}\left(\mathrm{JacobianMatrix}\left(\mathrm{sys}\,\mathrm{dec}\left[i\right]\,R\right)\,i=1..\mathrm{nops}\left(\mathrm{dec}\right)\right)$

$\left[\begin{array}{ccc}1& 1& 2z\\ 1& 2z& 1\\ 2z& 1& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 2\\ 1& 0& 1\\ 0& 1& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ 1& 2& 1\\ 0& 1& 1\end{array}\right],\left[\begin{array}{ccc}1& 1& 0\\ 1& 0& 1\\ 2& 1& 1\end{array}\right]$

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

$\mathrm{seq}\left(\mathrm{Determinant}\left(\mathrm{JacobianMatrix}\left(\mathrm{sys}\,\mathrm{dec}\left[i\right]\,R\right)\right)\,i=1..\mathrm{nops}\left(\mathrm{dec}\right)\right)$

$-8z^3+6z-2,0,0,0$