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RegularChains[MatrixTools]

 JacobianMatrix
 Jacobian matrix of a polynomial system modulo a regular chain

 Calling Sequence JacobianMatrix(sys, rc, R)

Parameters

 sys - system of polynomials in R rc - strongly normalized regular chain R - polynomial ring

Description

 • 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](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{MatrixTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,z\right]\right)$
 ${R}{:=}{\mathrm{polynomial_ring}}$ (1)
 > $\mathrm{sys}≔\left\{{x}^{2}+y+z-1,x+{y}^{2}+z-1,x+y+{z}^{2}-1\right\}$
 ${\mathrm{sys}}{:=}\left\{{{z}}^{{2}}{+}{x}{+}{y}{-}{1}{,}{{y}}^{{2}}{+}{x}{+}{z}{-}{1}{,}{{x}}^{{2}}{+}{y}{+}{z}{-}{1}\right\}$ (2)
 > $\mathrm{dec}≔\mathrm{Triangularize}\left(\mathrm{sys},R\right)$
 ${\mathrm{dec}}{:=}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (3)
 > $\mathrm{map}\left(\mathrm{Equations},\mathrm{dec},R\right)$
 $\left[\left[{x}{-}{z}{,}{y}{-}{z}{,}{{z}}^{{2}}{+}{2}{}{z}{-}{1}\right]{,}\left[{x}{,}{y}{,}{z}{-}{1}\right]{,}\left[{x}{,}{y}{-}{1}{,}{z}\right]{,}\left[{x}{-}{1}{,}{y}{,}{z}\right]\right]$
 $\left\{\left[{x}{,}{y}{,}{z}{-}{1}\right]{,}\left[{x}{,}{y}{-}{1}{,}{z}\right]{,}\left[{x}{-}{1}{,}{y}{,}{z}\right]{,}\left[{x}{-}{z}{,}{y}{-}{z}{,}{{z}}^{{2}}{+}{2}{}{z}{-}{1}\right]\right\}$ (4)
 > $\mathrm{seq}\left(\mathrm{JacobianMatrix}\left(\mathrm{sys},{\mathrm{dec}}_{i},R\right),i=1..\mathrm{nops}\left(\mathrm{dec}\right)\right)$
 $\left[\begin{array}{ccc}{1}& {1}& {2}{}{z}\\ {1}& {2}{}{z}& {1}\\ {2}{}{z}& {1}& {1}\end{array}\right]{,}\left[\begin{array}{rrr}{1}& {1}& {2}\\ {1}& {0}& {1}\\ {0}& {1}& {1}\end{array}\right]{,}\left[\begin{array}{rrr}{1}& {1}& {0}\\ {1}& {2}& {1}\\ {0}& {1}& {1}\end{array}\right]{,}\left[\begin{array}{rrr}{1}& {1}& {0}\\ {1}& {0}& {1}\\ {2}& {1}& {1}\end{array}\right]$
 $\left\{\left[\begin{array}{ccc}{1}& {1}& {2}{}{z}\\ {1}& {2}{}{z}& {1}\\ {2}{}{z}& {1}& {1}\end{array}\right]{,}\left[\begin{array}{rrr}{1}& {1}& {2}\\ {1}& {0}& {1}\\ {0}& {1}& {1}\end{array}\right]{,}\left[\begin{array}{rrr}{1}& {1}& {0}\\ {1}& {2}& {1}\\ {0}& {1}& {1}\end{array}\right]{,}\left[\begin{array}{rrr}{1}& {1}& {0}\\ {1}& {0}& {1}\\ {2}& {1}& {1}\end{array}\right]\right\}$ (5)
 > $\mathrm{with}\left(\mathrm{LinearAlgebra}\right):$
 > $\mathrm{seq}\left(\mathrm{Determinant}\left(\mathrm{JacobianMatrix}\left(\mathrm{sys},{\mathrm{dec}}_{i},R\right)\right),i=1..\mathrm{nops}\left(\mathrm{dec}\right)\right)$
 ${-}{8}{}{{z}}^{{3}}{+}{6}{}{z}{-}{2}{,}{0}{,}{0}{,}{0}$ (6)
 > $\left\{\mathrm{seq}\left(\mathrm{Determinant}\left(\mathrm{JacobianMatrix}\left(\mathrm{sys},{\mathrm{dec}}_{i},R\right)\right),i=1..\mathrm{nops}\left(\mathrm{dec}\right)\right)\right\}$
 $\left\{{0}{,}{-}{8}{}{{z}}^{{3}}{+}{6}{}{z}{-}{2}\right\}$ (7)