compute the intersection of two constructible sets
compute the intersection of two semi-algebraic sets
Intersection(cs1, cs2, R)
Intersection(lrsas1, lrsas2, R)
lists of regular semi-algebraic systems
This command computes the set-theoretic intersection of two constructible sets, or two semi-algebraic set, depending on the input type of its arguments.
A constructible set must be encoded as an constructible_set object, see the type definition in ConstructibleSetTools.
A semi-algebraic set must be encoded by a list of regular_semi_algebraic_system, see the type definition in RealTriangularize.
The command Intersection(cs1, cs2, R) returns the intersection of two constructible sets. The polynomial ring may have characteristic zero or a prime characteristic.
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.
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.
First, define the polynomial ring R and two polynomials of R.
R ≔ PolynomialRing⁡x,y,t
p ≔ 5⁢t+5⁢x−y−10⁢t+7
q ≔ 5⁢t−5⁢x−t+2⁢y−7⁢t+11
Using the GeneralConstruct command and adding one inequality, you can build a constructible set. Using the polynomials x−t and x+t for defining inequations, the two constructible sets cs1 and cs2 are different.
cs1 ≔ GeneralConstruct⁡p,q,x−t,R
cs2 ≔ GeneralConstruct⁡p,q,x+t,R
The intersection of cs1 and cs2 is a new constructible set cs.
cs ≔ Intersection⁡cs1,cs2,R
Check the result in another way.
cs3 ≔ GeneralConstruct⁡p,q,x+t,x−t,R
The results are as desired.
Consider now the semi-algebraic case:
lrsas1 ≔ RealTriangularize⁡p,q,,,x−t,R
lrsas2 ≔ RealTriangularize⁡p,q,,,x+t,R
lrsas ≔ Intersection⁡lrsas1,lrsas2,R
lrsas12 ≔ RealTriangularize⁡p,q,,,x+t,x−t,R
Verify the results
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.
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.
The RegularChains[SemiAlgebraicSetTools][Intersection] command was introduced in Maple 16.
The lrsas1 parameter was introduced in Maple 16.
For more information on Maple 16 changes, see Updates in Maple 16.
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