The command SeparateSolutions(l_rc, R) returns a list of square-free regular chains such that the ideals they generate are pairwise relatively prime.

The input regular chains must be zero-dimensional.

The algorithm is based on GCD computations.

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

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

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

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

$R≔\mathrm{polynomial\_ring}$

$\mathrm{rc1}≔\mathrm{Chain}\left(\left[x\,y\left(y+1\right)\right]\,\mathrm{Empty}\left(R\right)\,R\right)$

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

$\mathrm{rc2}≔\mathrm{Chain}\left(\left[x\,y\left(y+2\right)\right]\,\mathrm{Empty}\left(R\right)\,R\right)$

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

$\mathrm{out}≔\mathrm{SeparateSolutions}\left(\left[\mathrm{rc1}\,\mathrm{rc2}\right]\,R\right)$

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

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

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