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RegularChains[ConstructibleSetTools][Difference] - compute the difference of two constructible sets
RegularChains[SemiAlgebraicSetTools][Difference] - compute the difference of two semi-algebraic sets
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Calling Sequence
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Difference(cs1, cs2, R)
Difference(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
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list of regular semi-algebraic systems
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lrsas2
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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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This command computes the set-theoretic difference of two constructible sets or two semi-algebraic sets, depending on the input type.
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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 of RealTriangularize.
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The command Difference(cs1, cs2, R) returns a constructible set which is the set theoretic difference of two constructible sets. The polynomial ring may have characteristic zero or a prime characteristic.
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The command Difference(lrsas1, lrsas2, R) returns a list of regular semi-algebraic systems, encoding the set-theoretic difference of two semi-algebraic sets. The polynomial ring must have characteristic zero.
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The output may contain certain redundancy among the defining regular systems.
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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:-Difference or RegularChains:-SemiAlgebraicSetTools:-Difference.
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Compatibility
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The RegularChains[SemiAlgebraicSetTools][Difference] command was introduced in Maple 16.
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The lrsas1 and lrsas2 parameters were introduced in Maple 16.
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Examples
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First, define a polynomial ring.
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![R := PolynomialRing([x, y])](/support/helpjp/helpview.aspx?si=6387/file06470/math159.png)
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Use Triangularize to solve the polynomial equations given by  and  below.
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![dec := Triangularize([F, G], R, output = lazard)](/support/helpjp/helpview.aspx?si=6387/file06470/math188.png)
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![dec := [regular_chain, regular_chain, regular_chain, regular_chain, regular_chain, regular_chain]](/support/helpjp/helpview.aspx?si=6387/file06470/math191.png)
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![[[x-2, y+1], [x-1, y-1], [x+1, y-1], [x, y-3], [x-1, y+1], [x, y]]](/support/helpjp/helpview.aspx?si=6387/file06470/math198.png)
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Now remove points which cancel the following  .
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First, build the common solution set of  and  as a constructible set and also build one with  .
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![rcH := Chain([H], Empty(R), R)](/support/helpjp/helpview.aspx?si=6387/file06470/math221.png)
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![cs1 := ConstructibleSet([RegularSystem(rcH, R)], R)](/support/helpjp/helpview.aspx?si=6387/file06470/math228.png)
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Then use Difference to find a new constructible set which encodes those points that cancel  and  , but do not cancel  .
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You can check the result using the Info command: cs2 consists of six points, and five of them cancel  .
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![[[x-2, y+1], [1]], [[x-1, y-1], [1]], [[x+1, y-1], [1]], [[x, y-3], [1]], [[x-1, y+1], [1]], [[x, y], [1]]](/support/helpjp/helpview.aspx?si=6387/file06470/math271.png)
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The set cs3 consists of a single point which does not cancel  .
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![[[x, y-3], [1]]](/support/helpjp/helpview.aspx?si=6387/file06470/math284.png)
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An example on semi-algebraic set difference: to check whether or not two formula/conditions are equivalent. # Semi-algebraic Difference, verifying results
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![C1 := [0 < a, 0 < b, 0 < c, a < b+c, b < a+c, c < a+b, b^2+a^2-c^2 <= 0]](/support/helpjp/helpview.aspx?si=6387/file06470/math290.png)
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![C2 := [0 < a, 0 < b, 0 < c, a < b+c, b < a+c, c < a+b, c*(b^2+a^2-c^2)^2 < a*b^2*(2*a*c-c^2-a^2+b^2)]](/support/helpjp/helpview.aspx?si=6387/file06470/math294.png)
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![S1 := [C1, C2]](/support/helpjp/helpview.aspx?si=6387/file06470/math298.png)
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![S1 := [[0 < a, 0 < b, 0 < c, a < b+c, b < a+c, c < a+b, b^2+a^2-c^2 <= 0], [0 < a, 0 < b, 0 < c, a < b+c, b < a+c, c < a+b, c*(b^2+a^2-c^2)^2 < a*b^2*(2*a*c-c^2-a^2+b^2)]]](/support/helpjp/helpview.aspx?si=6387/file06470/math301.png)
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![S2 := [a-c < 0, 0 < a, 0 < b, 0 < c, a < b+c, b < a+c, c < a+b]](/support/helpjp/helpview.aspx?si=6387/file06470/math305.png)
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![S2 := [a-c < 0, 0 < a, 0 < b, 0 < c, a < b+c, b < a+c, c < a+b]](/support/helpjp/helpview.aspx?si=6387/file06470/math308.png)
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![R := PolynomialRing([a, b, c])](/support/helpjp/helpview.aspx?si=6387/file06470/math312.png)
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![dec1 := [regular_semi_algebraic_system, regular_semi_algebraic_system, regular_semi_algebraic_system]](/support/helpjp/helpview.aspx?si=6387/file06470/math319.png)
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![dec2 := [regular_semi_algebraic_system]](/support/helpjp/helpview.aspx?si=6387/file06470/math326.png)
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Verify dec1 = dec2 as set of points by Difference.
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![[]](/support/helpjp/helpview.aspx?si=6387/file06470/math341.png)
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![[]](/support/helpjp/helpview.aspx?si=6387/file06470/math348.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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