RegularChains[ParametricSystemTools] - Maple Programming Help

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RegularChains[ParametricSystemTools]

 DefiningSet
 compute the defining set of a regular chain

 Calling Sequence DefiningSet(rc, d, R)

Parameters

 rc - regular chain d - number of parameters R - polynomial ring

Description

 • 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](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right):$
 > $\mathrm{with}\left(\mathrm{ParametricSystemTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,u,v\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)

Consider the following parametric polynomial system F.

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

For different values of u and v, the solution set has a different nature. For example, u=0 and v=0 is a degenerate case: x=0 and y can be any value. To understand more about F, first decompose F into a set of regular chains.

 > $\mathrm{dec}≔\mathrm{Triangularize}\left(F,R,\mathrm{output}=\mathrm{lazard}\right)$
 ${\mathrm{dec}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (3)
 > $\mathrm{map}\left(\mathrm{Info},\mathrm{dec},R\right)$
 $\left[\left[\left({v}{}{y}{+}{1}\right){}{x}{-}{{y}}^{{2}}{}{{u}}^{{2}}{,}\left({{u}}^{{3}}{+}{{v}}^{{2}}\right){}{{y}}^{{2}}{+}{2}{}{v}{}{y}{+}{1}\right]{,}\left[{x}{,}{y}\right]{,}\left[{x}{,}{u}\right]{,}\left[\left({v}{}{y}{+}{1}\right){}{x}{-}{{y}}^{{2}}{}{{u}}^{{2}}{,}{2}{}{v}{}{y}{+}{1}{,}{{u}}^{{3}}{+}{{v}}^{{2}}\right]\right]$
 $\left[\left[\left({v}{}{y}{+}{1}\right){}{x}{-}{{y}}^{{2}}{}{{u}}^{{2}}{,}\left({{u}}^{{3}}{+}{{v}}^{{2}}\right){}{{y}}^{{2}}{+}{2}{}{v}{}{y}{+}{1}\right]{,}\left[{x}{,}{y}\right]{,}\left[{x}{,}{u}\right]{,}\left[\left({v}{}{y}{+}{1}\right){}{x}{-}{{y}}^{{2}}{}{{u}}^{{2}}{,}{2}{}{v}{}{y}{+}{1}{,}{{u}}^{{3}}{+}{{v}}^{{2}}\right]\right]$ (4)

The first regular chain is simple. For all values of u and v, it is well-specialized.

 > $\mathrm{ds1}≔\mathrm{DefiningSet}\left({\mathrm{dec}}_{1},2,R\right);$$\mathrm{Info}\left(\mathrm{ds1},R\right)$
 ${\mathrm{ds1}}{≔}{\mathrm{constructible_set}}$
 $\left[\left[{}\right]{,}\left[{u}{,}{{u}}^{{3}}{+}{{v}}^{{2}}\right]\right]$ (5)

For the last one, its defining set is given by ${u}^{3}+{v}^{2}=0$ and $v\ne 0$,  and the inequality is to ensure that rc1 specializes well.

 > $\mathrm{ds4}≔\mathrm{DefiningSet}\left({\mathrm{dec}}_{4},2,R\right);$$\mathrm{Info}\left(\mathrm{ds4},R\right)$
 ${\mathrm{ds4}}{≔}{\mathrm{constructible_set}}$
 $\left[\left[{{u}}^{{3}}{+}{{v}}^{{2}}\right]{,}\left[{v}\right]\right]$
 $\left\{\left[\left[{{u}}^{{3}}{+}{{v}}^{{2}}\right]{,}\left[{v}\right]\right]\right\}$ (6)