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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
Calling Sequence
Complement(cs, R)
Complement(lrsas, R)
Parameters
cs
-
constructible set
lrsas
list of regular semi-algebraic systems
R
polynomial ring
Description
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.
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.
The empty constructible set represents the empty set of .
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].
Compatibility
The RegularChains[SemiAlgebraicSetTools][Complement] command was introduced in Maple 16.
The lrsas 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 .
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.
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.
If you call Complement twice, you should retrieve the constructible set cs.
Semi-algebraic case
Verify compl = expected as set of points by Difference.
See Also
ConstructibleSet, ConstructibleSetTools, Difference, Intersection, Projection, RealTriangularize, RegularChains, SemiAlgebraicSetTools, Union
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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