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RegularChains[ConstructibleSetTools][RegularSystem] - construct a regular system from a regular chain and a list of inequations
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Calling Sequence
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RegularSystem(rc, H, R)
RegularSystem(rc, R)
RegularSystem(H, R)
RegularSystem(R)
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Parameters
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rc
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regular chain
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H
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list of polynomials of R
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R
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polynomial ring
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Description
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The command RegularSystem(rc, H, R) constructs a regular system from a regular chain and a list of inequations. Denote by  the quasi-component of rc. Then the constructed regular system encodes those points in  that do not cancel any polynomial in H.
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Each polynomial in H must be regular with respect to the regular chain rc; otherwise an error is reported.
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If rc is not specified, then rc is set to the empty regular chain.
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If H is not specified, then H is set to ![[1]](/support/helpjp/helpview.aspx?si=6414/file06484/math104.png) .
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The command RegularSystem(R) constructs the regular system corresponding to the whole space.
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This command is part of the RegularChains[ConstructibleSetTools] package, so it can be used in the form RegularSystem(..) 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][RegularSystem](..).
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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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Define a polynomial ring.
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![R := PolynomialRing([x, y, z])](/support/helpjp/helpview.aspx?si=6414/file06484/math149.png)
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Define a set of polynomials of R.
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![sys := [z*x^2+y+z, y^2+z]](/support/helpjp/helpview.aspx?si=6414/file06484/math160.png)
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![sys := [z*x^2+y+z, y^2+z]](/support/helpjp/helpview.aspx?si=6414/file06484/math163.png)
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![dec := [regular_chain, regular_chain]](/support/helpjp/helpview.aspx?si=6414/file06484/math170.png)
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There are two groups of solutions, each of which is given by a regular chain. To view the equations, use the Equations command.
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![[[z*x^2+y+z, y^2+z], [y, z]]](/support/helpjp/helpview.aspx?si=6414/file06484/math182.png)
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Let rc1 be the first regular chain, and rc2 be the second one.
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![rc1, rc2 := dec[1], dec[2]](/support/helpjp/helpview.aspx?si=6414/file06484/math192.png)
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Consider two polynomials h1 and h2; regard them as inequations.
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To obtain regular systems, first check if  is regular with respect to  , and  is regular with respect to  .
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Both of them are regular, thus you can build the following regular systems.
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![rs1 := RegularSystem(rc1, [h1], R)](/support/helpjp/helpview.aspx?si=6414/file06484/math235.png) ![rs2 := RegularSystem(rc2, [h2], R)](/support/helpjp/helpview.aspx?si=6414/file06484/math236.png)
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You can simply call RegularSystem(R) to build the regular system which encodes all points.
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![ws := ConstructibleSet([RegularSystem(R)], R)](/support/helpjp/helpview.aspx?si=6414/file06484/math250.png)
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The complement of  must be empty.
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| (10) |
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