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RegularChains[ConstructibleSetTools][ConstructibleSet] - construct a constructible set from a list or set of regular systems
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Calling Sequence
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ConstructibleSet(lrs, R)
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Parameters
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lrs
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list or set of regular systems
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R
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polynomial ring
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Description
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The command ConstructibleSet(lrs, R) returns a constructible set defined by the list lrs of regular systems.
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A point belongs to a constructible set if and only if it is a solution of one of its defining regular systems. That is, a constructible set is the union of the solution sets of its defining regular systems.
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Since a regular system always defines a nonempty set, a constructible set is empty if and only if its list of defining regular systems is empty.
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This command is part of the RegularChains[ConstructibleSetTools] package, so it can be used in the form ConstructibleSet(..) 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][ConstructibleSet](..).
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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.
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Examples
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This example demonstrates how to build a constructible set structure.
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First, define a polynomial ring.
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![R := PolynomialRing([x, y, a, b, c, d])](/support/helpjp/helpview.aspx?si=6411/file06468/math94.png)
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Consider the following linear polynomial system.
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![sys := [a*x+b*y, c*x+d*y]](/support/helpjp/helpview.aspx?si=6411/file06468/math103.png)
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![sys := [a*x+b*y, c*x+d*y]](/support/helpjp/helpview.aspx?si=6411/file06468/math106.png)
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The command Triangularize with lazard option decomposes the solution set by means of regular chains. Each regular chain describes a group of solutions with certain mathematical meaning. See RegularChains for more information.
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![dec := [regular_chain, regular_chain, regular_chain, regular_chain, regular_chain, regular_chain]](/support/helpjp/helpview.aspx?si=6411/file06468/math122.png)
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To build constructible sets, you first need to create regular systems. For simplicity, just let ![[1]](/support/helpjp/helpview.aspx?si=6411/file06468/math125.png) be the inequation part of each regular system.
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![lrs := map(RegularSystem, dec, [1], R)](/support/helpjp/helpview.aspx?si=6411/file06468/math130.png)
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![lrs := [regular_system, regular_system, regular_system, regular_system, regular_system, regular_system]](/support/helpjp/helpview.aspx?si=6411/file06468/math133.png)
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Then  is a list of regular systems by which you can create a constructible set cs.
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Use Info to see its internal defining polynomials.
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![[[x, y], [1]], [[c*x+y*d, d*a-b*c], [1]], [[y, a, c], [1]], [[x, b, d], [1]], [[a*x+y*b, c, d], [1]], [[a, b, c, d], [1]]](/support/helpjp/helpview.aspx?si=6411/file06468/math157.png)
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