The command RepresentingChain(rs, R) returns the representing regular chain of the regular system rs, where the polynomials of rs belong to R.

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

See ConstructibleSetTools and RegularChains for the related mathematical concepts, in particular for the ideas of a constructible set, a regular system, and a regular chain.

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

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

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

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

$R≔\mathrm{polynomial\_ring}$

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

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

$\mathrm{dec}≔\mathrm{Triangularize}\left(\mathrm{sys}\,R\,\mathrm{output}=\mathrm{lazard}\right)$

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

$\mathrm{rc}≔\mathrm{dec}\left[1\right]$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$h≔x+z$

$h≔x+z$

$\mathrm{IsRegular}\left(h\,\mathrm{rc}\,R\right)$

$\mathrm{true}$

$\mathrm{rs}≔\mathrm{RegularSystem}\left(\mathrm{rc}\,\left[h\right]\,R\right)$

$\mathrm{rs}≔\mathrm{regular\_system}$

$\mathrm{EqualSaturatedIdeals}\left(\mathrm{rc}\,\mathrm{RepresentingChain}\left(\mathrm{rs}\,R\right)\,R\right)$

$\mathrm{true}$