The function call MatrixCombine(lrc, R, lm) returns the equiprojectable decomposition of the variety given by lrc, and the corresponding combined matrices.

The variety encoded by lrc is the union of the regular zero sets of the regular chains of lrc.

It is assumed that every regular chain in lrc is zero-dimensional and strongly normalized, and that all matrices in lm have the same format.

It is also assumed that lrc and lm have the same number of elements.

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

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

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

$\mathrm{rc}≔\mathrm{Empty}\left(R\right)\:$

$\mathrm{rc1}≔\mathrm{Chain}\left(\left[z\,y\,x-1\right]\,\mathrm{rc}\,R\right)\:$

$\mathrm{rc2}≔\mathrm{Chain}\left(\left[z\,y-1\,x\right]\,\mathrm{rc}\,R\right)\:$

$\mathrm{rc3}≔\mathrm{Chain}\left(\left[z-1\,y\,x\right]\,\mathrm{rc}\,R\right)\:$

$\mathrm{rc4}≔\mathrm{Chain}\left(\left[z^2+2z-1\,y-z\,x-z\right]\,\mathrm{rc}\,R\right)\:$

$\mathrm{m1}≔\mathrm{Matrix}\left(\left[\left[x+y-z\,x-y+1\right]\,\left[x-2y+1\,x-2+z\right]\right]\right)$

$\mathrm{m1}≔\left[\begin{array}{cc}x+y-z& x-y+1\\ x-2y+1& x-2+z\end{array}\right]$

$\mathrm{m2}≔\mathrm{Matrix}\left(\left[\left[x-z-1\,x+z-2\right]\,\left[x-3+y\,x+y+z-1\right]\right]\right)$

$\mathrm{m2}≔\left[\begin{array}{cc}x-z-1& x-2+z\\ x-3+y& x+y+z-1\end{array}\right]$

$\mathrm{m3}≔\mathrm{Matrix}\left(\left[\left[x-y-1\,x+y-2z\right]\,\left[x+y-z-3\,x+y+z-2\right]\right]\right)$

$\mathrm{m3}≔\left[\begin{array}{cc}x-y-1& x+y-2z\\ x+y-z-3& x+y+z-2\end{array}\right]$

$\mathrm{m4}≔\mathrm{Matrix}\left(\left[\left[x-z-y+1\,x+2z-2y\right]\,\left[x+y-z+1\,x+y-z-1\right]\right]\right)$

$\mathrm{m4}≔\left[\begin{array}{cc}x-z-y+1& x+2z-2y\\ x+y-z+1& x+y-z-1\end{array}\right]$

$\mathrm{clr}≔\mathrm{MatrixCombine}\left(\left[\mathrm{rc1}\,\mathrm{rc2}\,\mathrm{rc3}\,\mathrm{rc4}\right]\,R\,\left[\mathrm{m1}\,\mathrm{m2}\,\mathrm{m3}\,\mathrm{m4}\right]\right)$

$\mathrm{clr}≔\left[\left[\left[\begin{array}{cc}-2y+1& -4y+2\\ -4y+2& y-1\end{array}\right]\,\mathrm{regular\_chain}\right]\,\left[\left[\begin{array}{cc}-\frac{1}{2}{z}^2-2z+\frac{3}{2}& -\frac{3}{2}{z}^2-2z+\frac{3}{2}\\ -3z^2-5z+4& -\frac{z^2}{2}-\frac{1}{2}\end{array}\right]\,\mathrm{regular\_chain}\right]\right]$

$\mathrm{rc1}≔\mathrm{clr}\left[1\right]\left[2\right]\;$$\mathrm{Equations}\left(\mathrm{rc1}\,R\right)$

$\mathrm{rc1}≔\mathrm{regular\_chain}$

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

$\mathrm{rc2}≔\mathrm{clr}\left[2\right]\left[2\right]\;$$\mathrm{Equations}\left(\mathrm{rc2}\,R\right)$

$\mathrm{rc2}≔\mathrm{regular\_chain}$

$\left[2x+z^2-1\,2y+z^2-1\,z^3+z^2-3z+1\right]$