The command LowerEchelonForm(A, rc, R) returns a list of pairs $\left[B_i\,{\mathrm{rc}}_i\right]$ where ${\mathrm{rc}}_i$ is a regular chain, and $B_i$ is the lower echelon form of A modulo the saturated ideal of rc_i.

All the returned regular chains ${\mathrm{rc}}_i$ form a triangular decomposition of rc (in the sense of Kalkbrener).

It is assumed that rc is strongly normalized.

The algorithm is an adaptation of the algorithm of Bareiss.

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

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

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

$R≔\mathrm{polynomial\_ring}$

$T≔\mathrm{Empty}\left(R\right)\:$

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

$T≔\mathrm{regular\_chain}$

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

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

$m≔\mathrm{Matrix}\left(\left[\left[x\,y\,z\right]\,\left[x+1\,y+2\,z+3\right]\,\left[x+4\,y+5\,z+6\right]\right]\right)$

$m≔\left[\begin{array}{ccc}x& y& z\\ x+1& y+2& z+3\\ x+4& y+5& z+6\end{array}\right]$

$\mathrm{lem}≔\mathrm{LowerEchelonForm}\left(m\,T\,R\right)$

$\mathrm{lem}≔\left[\left[\left[\begin{array}{ccc}6& 0& 0\\ 0& 3& 0\\ x+4& y+5& z+6\end{array}\right]\,\mathrm{regular\_chain}\right]\,\left[\left[\begin{array}{ccc}12& 0& 0\\ \mathrm{-6}& 3& 0\\ x+4& y+5& z+6\end{array}\right]\,\mathrm{regular\_chain}\right]\,\left[\left[\begin{array}{ccc}0& 0& 0\\ \mathrm{-6}& \mathrm{-3}& 0\\ x+4& y+5& z+6\end{array}\right]\,\mathrm{regular\_chain}\right]\,\left[\left[\begin{array}{ccc}-3x+6y+6& 0& 0\\ 3x& 3y+3& 0\\ x+4& y+5& z+6\end{array}\right]\,\mathrm{regular\_chain}\right]\right]$