The command EquiprojectableDecomposition(lrc, R) returns the equiprojectable decomposition of the variety given by lrc.

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.

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

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

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

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

$R≔\mathrm{polynomial\_ring}$

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

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

$\mathrm{lrc}≔\mathrm{Triangularize}\left(\mathrm{sys}\,R\,\mathrm{normalized}=\mathrm{yes}\right)$

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

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

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

$\mathrm{ed}≔\mathrm{EquiprojectableDecomposition}\left(\mathrm{lrc}\,R\right)$

$\mathrm{ed}≔\left\{\mathrm{regular\_chain}\,\mathrm{regular\_chain}\right\}$

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

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

Dahan, X.; Moreno Maza, M.; Schost, E.; Wu, W. and Xie, Y. "Equiprojectable decompositions of zero-dimensional varieties" In proc. of International Conference on Polynomial System Solving, University of Paris 6, France, 2004.