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RegularChains[ConstructibleSetTools][Intersection] - compute the intersection of two constructible sets
RegularChains[SemiAlgebraicSetTools][Intersection] - compute the intersection of two semi-algebraic sets
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Calling Sequence
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Intersection(cs1, cs2, R)
Intersection(lrsas1, lrsas2, R)
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Parameters
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cs1, cs2
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constructible sets
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lrsas1, lrsas2
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lists 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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This command computes the set-theoretic intersection of two constructible sets, or two semi-algebraic set, depending on the input type of its arguments.
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A constructible set must be encoded as an constructible_set object, see the type definition in ConstructibleSetTools.
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A semi-algebraic set must be encoded by a list of regular_semi_algebraic_system, see the type definition in RealTriangularize.
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The command Intersection(cs1, cs2, R) returns the intersection of two constructible sets. The polynomial ring may have characteristic zero or a prime characteristic.
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The command Intersection(lrsas1, lrsas2, R) returns the intersection of two semi-algebraic sets, encoded by list of regular_semi_algebraic_system. The polynomial ring must have characteristic zero.
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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:-Intersection or RegularChains:-SemiAlgebraicSetTools:-Intersection.
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Compatibility
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The RegularChains[SemiAlgebraicSetTools][Intersection] command was introduced in Maple 16.
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The lrsas1 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, y, t])](/support/helpjp/helpview.aspx?si=6388/file06473/math147.png)
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Using the GeneralConstruct command and adding one inequality, you can build a constructible set. Using the polynomials  and  for defining inequations, the two constructible sets cs1 and cs2 are different.
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![cs1 := GeneralConstruct([p, q], [x-t], R)](/support/helpjp/helpview.aspx?si=6388/file06473/math180.png)
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![cs2 := GeneralConstruct([p, q], [x+t], R)](/support/helpjp/helpview.aspx?si=6388/file06473/math187.png)
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The intersection of cs1 and cs2 is a new constructible set cs.
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Check the result in another way.
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![cs3 := GeneralConstruct([p, q], [x+t, x-t], R)](/support/helpjp/helpview.aspx?si=6388/file06473/math211.png)
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The results are as desired.
Consider now the semi-algebraic case:
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![lrsas1 := RealTriangularize([p, q], [], [], [x-t], R)](/support/helpjp/helpview.aspx?si=6388/file06473/math229.png)
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![lrsas1 := [regular_semi_algebraic_system, regular_semi_algebraic_system]](/support/helpjp/helpview.aspx?si=6388/file06473/math232.png)
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![lrsas2 := RealTriangularize([p, q], [], [], [x+t], R)](/support/helpjp/helpview.aspx?si=6388/file06473/math236.png)
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![lrsas2 := [regular_semi_algebraic_system, regular_semi_algebraic_system]](/support/helpjp/helpview.aspx?si=6388/file06473/math239.png)
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![lrsas := [regular_semi_algebraic_system, regular_semi_algebraic_system]](/support/helpjp/helpview.aspx?si=6388/file06473/math246.png)
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![lrsas12 := RealTriangularize([p, q], [], [], [x+t, x-t], R)](/support/helpjp/helpview.aspx?si=6388/file06473/math250.png)
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![lrsas12 := [regular_semi_algebraic_system, regular_semi_algebraic_system]](/support/helpjp/helpview.aspx?si=6388/file06473/math253.png)
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Verify the results
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![[]](/support/helpjp/helpview.aspx?si=6388/file06473/math262.png)
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![[]](/support/helpjp/helpview.aspx?si=6388/file06473/math269.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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