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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
Calling Sequence
Intersection(cs1, cs2, R)
Intersection(lrsas1, lrsas2, R)
Parameters
cs1, cs2
-
constructible sets
lrsas1, lrsas2
lists of regular semi-algebraic systems
R
polynomial ring
Description
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.
Compatibility
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.
Examples
First, define the polynomial ring and two polynomials of .
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.
The intersection of cs1 and cs2 is a new constructible set cs.
Check the result in another way.
The results are as desired.
Consider now the semi-algebraic case:
Verify the results
See Also
Complement, ConstructibleSet, ConstructibleSetTools, Difference, GeneralConstruct, RealTriangularize, RegularChains, RegularChains, SemiAlgebraicSetTools
References
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.
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