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RegularChains[ConstructibleSetTools][MakePairwiseDisjoint] - make the defining regular systems in a constructible set pairwise disjoint
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Calling Sequence
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MakePairwiseDisjoint(cs, R)
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Parameters
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cs
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constructible set
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R
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polynomial ring
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Description
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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.
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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.
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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](..).
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Examples
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First, define the polynomial ring.
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![R := PolynomialRing([x, y, a, b, c, d, e])](/support/helpjp/helpview.aspx?si=6405/file06476/math87.png)
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Consider the following almost general linear equations. They are not completely general, since their constant term, namely  , is the same.
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After projecting the variety defined by  and  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.
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![cs := Projection([F, G], 5, R)](/support/helpjp/helpview.aspx?si=6405/file06476/math118.png)
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![lrs := [regular_system, regular_system, regular_system, regular_system, regular_system, regular_system, regular_system, regular_system, regular_system]](/support/helpjp/helpview.aspx?si=6405/file06476/math128.png)
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![{[[], [c, d*a-b*c]], [[c], [d, a]], [[a-c, b-d], [c]], [[d*a-b*c, e], [d, c]], [[a, c, e], [1]], [[a, b-d, c], [d]], [[b, d, e], [1]], [[c, d, e], [a]], [[a, b, c, d, e], [1]]}](/support/helpjp/helpview.aspx?si=6405/file06476/math138.png)
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There are 9 regular systems defining the image cs of the projection. To remove common parts of these regular systems, use MakePairwiseDisjoint.
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![lcs_mpd := [regular_system, regular_system, regular_system, regular_system, regular_system, regular_system, regular_system, regular_system, regular_system]](/support/helpjp/helpview.aspx?si=6405/file06476/math165.png)
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Now, there are 10 components.
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![[[a, b, c, d, e], [1]], [[c, d, e], [a, b]], [[b, d, e], [a-c]], [[a, c, e], [b-d]], [[d*a-b*c, e], [d, c, b-d]], [[a, b-d, c], [d]], [[c], [d, a]], [[a-c, b-d], [c]], [[], [c, d*a-b*c]]](/support/helpjp/helpview.aspx?si=6405/file06476/math181.png)
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Notice that some components have split during the redundancy removal.
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