compute the complement of a constructible set
compute the complement of a semi-algebraic set
list of regular semi-algebraic systems
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 Kn. 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 Kn.
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].
First define the polynomial ring R and two polynomials of R.
R ≔ PolynomialRing⁡x,a,b,c,d
F ≔ a⁢x+c
G ≔ b⁢x+d
The goal is to determine for which parameter values of a, b, c and d the generic linear equations F and G have solutions. Project the variety defined by F and G onto the parameter space.
cs ≔ Projection⁡F,G,4,R
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.
com_cs ≔ Complement⁡cs,R
If you call Complement twice, you should retrieve the constructible set cs.
com_com_cs ≔ Complement⁡com_cs,R
R ≔ PolynomialRing⁡a,b,c:
S1 ≔ a2−c⁢a−a=0,0<a−c
out ≔ RealTriangularize⁡S1,R
compl ≔ Complement⁡out,R
expected ≔ a2−c⁢a−a≠0,a−c≤0
expected ≔ map⁡t→op⁡RealTriangularize⁡t,R,expected
Verify compl = expected as set of points by Difference.
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][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.
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