The command RepresentingRegularSystems(cs,R) returns a list of regular systems which defines the constructible set cs, that is, a list of regular systems (whose polynomials belong to R) such that the union of their zero sets is exactly equal to cs.

Recall that every constructible set built by the ConstructibleSetTools module is in fact represented by a list of regular systems representing it in the above sense.

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.

The command RepresentingRegularSystems is part of the RegularChains[ConstructibleSetTools] package, so it can be used in the form RepresentingRegularSystems(..) 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][RepresentingRegularSystems](..).

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

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

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

$R≔\mathrm{polynomial\_ring}$

$f≔ux+v\;$$g≔vy+u$

$f≔ux+v$

$g≔vy+u$

$\mathrm{cs}≔\mathrm{GeneralConstruct}\left(\left[f\,g\right]\,\left[u^2+v^2-1\right]\,R\right)$

$\mathrm{cs}≔\mathrm{constructible\_set}$

$\mathrm{lrs}≔\mathrm{RepresentingRegularSystems}\left(\mathrm{cs}\,R\right)$

$\mathrm{lrs}≔\left[\mathrm{regular\_system}\,\mathrm{regular\_system}\right]$

$\mathrm{lrc}≔\mathrm{map}\left(\mathrm{RepresentingChain}\,\mathrm{lrs}\,R\right)$

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

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

$\left[\left[ux+v\,vy+u\right]\,\left[u\,v\right]\right]$

$\mathrm{map}\left(\mathrm{RepresentingInequations}\,\mathrm{lrs}\,R\right)$

$\left[\left[u^2+v^2-1\right]\,\left[\right]\right]$