The command DefiningSet(rc, d, R) returns the defining set of rc with respect to the last d variables, regarded as parameters. This is a constructible set $C$.

Given a positive integer d, the regular chain rc can be split into two parts. Denote by $\mathrm{rc0}$ the set of the polynomials in rc involving only the last d variables, and denote by $\mathrm{rc1}$ the other polynomials of rc. Certainly, both $\mathrm{rc0}$ and $\mathrm{rc1}$ are regular chains.

Let $W$ be the quasi-component of $\mathrm{rc0}$. For a point $P$ in $W$, after specializing $\mathrm{rc1}$ at $P$, two situations arise:

(1) either $\mathrm{rc1}$ is not a regular chain anymore;

(2) or $\mathrm{rc1}$ is still a regular chain.

There is a subtle point: after specializing $\mathrm{rc1}$ at $P$, it might happen that it is still a regular chain, but its shape changes. In other words, the degree of the geometric object given by $\mathrm{rc1}$ could change. The term specialize well, defined below, takes these cases into account.

The regular chain rc specializes well at a point $P$ of $W$ if $\mathrm{rc1}$ is a regular chain after specialization and no initial of polynomials in rc1 vanish during the specialization.

The defining set of rc with respect to the last d variables consists of the points in $W$ at which rc specializes well.

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

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

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

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

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

$R≔\mathrm{polynomial\_ring}$

$F≔\left[vxy+u{x}^2+x\,u{y}^2+x^2\right]$

$F≔\left[u{x}^2+vxy+x\,u{y}^2+x^2\right]$

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

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

$\mathrm{map}\left(\mathrm{Info}\,\mathrm{dec}\,R\right)$

$\left[\left[\left(vy+1\right)x-y^2{u}^2\,\left(u^3+v^2\right)y^2+2vy+1\right]\,\left[x\,y\right]\,\left[x\,u\right]\,\left[\left(vy+1\right)x-y^2{u}^2\,2vy+1\,u^3+v^2\right]\right]$

$\mathrm{ds1}≔\mathrm{DefiningSet}\left(\mathrm{dec}\left[1\right]\,2\,R\right)\;$$\mathrm{Info}\left(\mathrm{ds1}\,R\right)$

$\mathrm{ds1}≔\mathrm{constructible\_set}$

$\left[\left[\right]\,\left[u\,u^3+v^2\right]\right]$

$\mathrm{ds4}≔\mathrm{DefiningSet}\left(\mathrm{dec}\left[4\right]\,2\,R\right)\;$$\mathrm{Info}\left(\mathrm{ds4}\,R\right)$

$\mathrm{ds4}≔\mathrm{constructible\_set}$

$\left[\left[u^3+v^2\right]\,\left[v\right]\right]$