RegularChains[ConstructibleSetTools] - Maple Programming Help

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RegularChains[ConstructibleSetTools]

 MakePairwiseDisjoint
 make the defining regular systems in a constructible set pairwise disjoint

 Calling Sequence MakePairwiseDisjoint(cs, R)

Parameters

 cs - constructible set R - polynomial ring

Description

 • The command MakePairwiseDisjoint(cs, R) returns a constructible set cs1 such that cs1 and cs are equal and the regular systems representing cs1 are pairwise disjoint.
 • Generally, in a constructible set, there is some redundancy among its components defined by regular systems. By default, functions on constructible sets do not remove redundancy because such a computation is expensive.
 • This command is part of the RegularChains[ConstructibleSetTools] package, so it can be used in the form MakePairwiseDisjoint(..) 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][MakePairwiseDisjoint](..).

Examples

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

First, define the polynomial ring.

 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,a,b,c,d,e\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)

Consider the following almost general linear equations. They are not completely general, since their constant term, namely $e$, is the same.

 > $F≔ax+by-e$
 ${F}{≔}{a}{}{x}{+}{b}{}{y}{-}{e}$ (2)
 > $G≔cx+dy-e$
 ${G}{≔}{c}{}{x}{+}{d}{}{y}{-}{e}$ (3)

After projecting the variety defined by $F$ and $G$ into the parameter space given by the last 5 variables, you can see when such general linear equations have solutions after specializing the last 5 variables.

 > $\mathrm{cs}≔\mathrm{Projection}\left(\left[F,G\right],5,R\right)$
 ${\mathrm{cs}}{≔}{\mathrm{constructible_set}}$ (4)
 > $\mathrm{lrs}≔\mathrm{RepresentingRegularSystems}\left(\mathrm{cs},R\right)$
 ${\mathrm{lrs}}{≔}\left[{\mathrm{regular_system}}{,}{\mathrm{regular_system}}{,}{\mathrm{regular_system}}{,}{\mathrm{regular_system}}{,}{\mathrm{regular_system}}{,}{\mathrm{regular_system}}{,}{\mathrm{regular_system}}{,}{\mathrm{regular_system}}{,}{\mathrm{regular_system}}\right]$ (5)
 > $\mathrm{Info}\left(\mathrm{cs},R\right)$
 $\left[\left[{}\right]{,}\left[{c}{,}{d}{}{a}{-}{b}{}{c}\right]\right]{,}\left[\left[{a}{-}{c}{,}{b}{-}{d}\right]{,}\left[{c}\right]\right]{,}\left[\left[{c}\right]{,}\left[{d}{,}{a}\right]\right]{,}\left[\left[{d}{}{a}{-}{b}{}{c}{,}{e}\right]{,}\left[{d}{,}{c}\right]\right]{,}\left[\left[{a}{,}{b}{-}{d}{,}{c}\right]{,}\left[{d}\right]\right]{,}\left[\left[{a}{,}{c}{,}{e}\right]{,}\left[{1}\right]\right]{,}\left[\left[{b}{,}{d}{,}{e}\right]{,}\left[{1}\right]\right]{,}\left[\left[{c}{,}{d}{,}{e}\right]{,}\left[{a}\right]\right]{,}\left[\left[{a}{,}{b}{,}{c}{,}{d}{,}{e}\right]{,}\left[{1}\right]\right]$
 $\left\{\left[\left[{}\right]{,}\left[{c}{,}{d}{}{a}{-}{b}{}{c}\right]\right]{,}\left[\left[{c}\right]{,}\left[{d}{,}{a}\right]\right]{,}\left[\left[{a}{-}{c}{,}{b}{-}{d}\right]{,}\left[{c}\right]\right]{,}\left[\left[{d}{}{a}{-}{b}{}{c}{,}{e}\right]{,}\left[{d}{,}{c}\right]\right]{,}\left[\left[{a}{,}{c}{,}{e}\right]{,}\left[{1}\right]\right]{,}\left[\left[{a}{,}{b}{-}{d}{,}{c}\right]{,}\left[{d}\right]\right]{,}\left[\left[{b}{,}{d}{,}{e}\right]{,}\left[{1}\right]\right]{,}\left[\left[{c}{,}{d}{,}{e}\right]{,}\left[{a}\right]\right]{,}\left[\left[{a}{,}{b}{,}{c}{,}{d}{,}{e}\right]{,}\left[{1}\right]\right]\right\}$ (6)
 > $\mathrm{nops}\left(\mathrm{lrs}\right)$
 ${9}$ (7)

There are 9 regular systems defining the image cs of the projection. To remove common parts of these regular systems, use MakePairwiseDisjoint.

 > $\mathrm{cs_mpd}≔\mathrm{MakePairwiseDisjoint}\left(\mathrm{cs},R\right)$
 ${\mathrm{cs_mpd}}{≔}{\mathrm{constructible_set}}$ (8)
 > $\mathrm{lcs_mpd}≔\mathrm{RepresentingRegularSystems}\left(\mathrm{cs_mpd},R\right)$
 ${\mathrm{lcs_mpd}}{≔}\left[{\mathrm{regular_system}}{,}{\mathrm{regular_system}}{,}{\mathrm{regular_system}}{,}{\mathrm{regular_system}}{,}{\mathrm{regular_system}}{,}{\mathrm{regular_system}}{,}{\mathrm{regular_system}}{,}{\mathrm{regular_system}}{,}{\mathrm{regular_system}}\right]$ (9)
 > $\mathrm{nops}\left(\mathrm{lcs_mpd}\right)$
 ${9}$ (10)

Now, there are 10 components.

 > $\mathrm{Info}\left(\mathrm{cs_mpd},R\right)$
 $\left[\left[{a}{,}{b}{,}{c}{,}{d}{,}{e}\right]{,}\left[{1}\right]\right]{,}\left[\left[{c}{,}{d}{,}{e}\right]{,}\left[{a}{,}{b}\right]\right]{,}\left[\left[{b}{,}{d}{,}{e}\right]{,}\left[{a}{-}{c}\right]\right]{,}\left[\left[{a}{,}{c}{,}{e}\right]{,}\left[{b}{-}{d}\right]\right]{,}\left[\left[{d}{}{a}{-}{b}{}{c}{,}{e}\right]{,}\left[{d}{,}{c}{,}{b}{-}{d}\right]\right]{,}\left[\left[{a}{,}{b}{-}{d}{,}{c}\right]{,}\left[{d}\right]\right]{,}\left[\left[{c}\right]{,}\left[{d}{,}{a}\right]\right]{,}\left[\left[{a}{-}{c}{,}{b}{-}{d}\right]{,}\left[{c}\right]\right]{,}\left[\left[{}\right]{,}\left[{c}{,}{d}{}{a}{-}{b}{}{c}\right]\right]$ (11)

Notice that some components have split during the redundancy removal.