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RegularChains[ConstructibleSetTools][Complement] - compute the complement of a constructible set
RegularChains[SemiAlgebraicSetTools][Complement] - compute the complement of a semi-algebraic set
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Calling Sequence
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Complement(cs, R)
Complement(lrsas, R)
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Parameters
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cs
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constructible set
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lrsas
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list of regular semi-algebraic systems
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R
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polynomial ring
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Description
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The command Complement(cs, R) returns the complement of the constructible set cs in the affine space associated with R. If K is the algebraic closure of the coefficient field of R and n is the number of variables in R, then this affine space is  . The polynomial ring may have characteristic zero or a prime characteristic.
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The command Complement(lrsas, R) returns the complement of the semi-algebraic set represented by lrsas (see RealTriangularize for this representation). The polynomial ring must have characteristic zero. The empty semi-algebraic set is encoded by the empty list.
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The empty constructible set represents the empty set of  .
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This command is available once RegularChains[ConstructibleSetTools] submodule or RegularChains[SemiAlgebraicSetTools] submodule have been loaded. It can always be accessed through one of the following long forms: RegularChains[ConstructibleSetTools][Complement] or RegularChains[SemiAlgebraicSetTools][Complement].
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Compatibility
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The RegularChains[SemiAlgebraicSetTools][Complement] command was introduced in Maple 16.
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The lrsas parameter was introduced in Maple 16.
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Examples
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First define the polynomial ring  and two polynomials of  .
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![R := PolynomialRing([x, a, b, c, d])](/support/helpjp/helpview.aspx?si=6386/file06467/math148.png)
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The goal is to determine for which parameter values of  ,  ,  and  the generic linear equations  and  have solutions. Project the variety defined by  and  onto the parameter space.
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![cs := Projection([F, G], 4, R)](/support/helpjp/helpview.aspx?si=6386/file06467/math187.png)
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![[[d*a-b*c], [d, b]], [[b, d], [a]], [[c, d], [1]], [[a, b, c, d], [1]]](/support/helpjp/helpview.aspx?si=6386/file06467/math197.png)
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Therefore, four regular systems encode this projection in the parameter space. The complement of cs should be those points that make the linear equations have no common solutions.
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![[[], [d*a-b*c, d]], [[d], [c, b]], [[a, b], [d]], [[a, b, d], [c]]](/support/helpjp/helpview.aspx?si=6386/file06467/math215.png)
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If you call Complement twice, you should retrieve the constructible set cs.
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Semi-algebraic case
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![R := PolynomialRing([a, b, c])](/support/helpjp/helpview.aspx?si=6386/file06467/math241.png)
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![S1 := [a^2-c*a-a = 0, 0 < a-c]](/support/helpjp/helpview.aspx?si=6386/file06467/math245.png)
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![S1 := [a^2-c*a-a = 0, 0 < a-c]](/support/helpjp/helpview.aspx?si=6386/file06467/math248.png)
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![out := [regular_semi_algebraic_system, regular_semi_algebraic_system]](/support/helpjp/helpview.aspx?si=6386/file06467/math255.png)
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![compl := [regular_semi_algebraic_system, regular_semi_algebraic_system, regular_semi_algebraic_system]](/support/helpjp/helpview.aspx?si=6386/file06467/math262.png)
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![expected := [[a^2-c*a-a <> 0], [a-c <= 0]]](/support/helpjp/helpview.aspx?si=6386/file06467/math266.png)
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![expected := [[a^2-c*a-a <> 0], [a-c <= 0]]](/support/helpjp/helpview.aspx?si=6386/file06467/math269.png)
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![expected := [regular_semi_algebraic_system, regular_semi_algebraic_system, regular_semi_algebraic_system]](/support/helpjp/helpview.aspx?si=6386/file06467/math276.png)
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Verify compl = expected as set of points by Difference.
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![[]](/support/helpjp/helpview.aspx?si=6386/file06467/math291.png)
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![[]](/support/helpjp/helpview.aspx?si=6386/file06467/math298.png)
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References
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Chen, C.; Golubitsky, O.; Lemaire, F.; Moreno Maza, M.; and Pan, W. "Comprehensive Triangular Decomposition". Proc. CASC 2007, LNCS, Vol. 4770: 73-101. Springer, 2007.
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Chen, C.; Davenport, J.-D.; Moreno Maza, M.; Xia, B.; and Xiao, R. "Computing with semi-algebraic sets represented by triangular decomposition". Proceedings of 2011 International Symposium on Symbolic and Algebraic Computation (ISSAC 2011), ACM Press, pp. 75--82, 2011.
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