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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 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].

Examples

withRegularChains:

withChainTools:

withConstructibleSetTools:

First define the polynomial ring R and two polynomials of R.

R:=PolynomialRingx,a,b,c,d

R:=polynomial_ring

(1)

F:=ax+c

F:=ax+c

(2)

G:=bx+d

G:=bx+d

(3)

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:=ProjectionF,G,4,R

cs:=constructible_set

(4)

Infocs,R

dabc,d,b,b,d,a,c,d,1,a,b,c,d,1

(5)

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:=Complementcs,R

com_cs:=constructible_set

(6)

Infocom_cs,R

,dabc,d,d,c,b,a,b,d,a,b,d,c

(7)

If you call Complement twice, you should retrieve the constructible set cs.

com_com_cs:=Complementcom_cs,R

com_com_cs:=constructible_set

(8)

IsContainedcs,com_com_cs,RandIsContainedcom_com_cs,cs,R

true

(9)

Semi-algebraic case

R:=PolynomialRinga,b,c:

S1:=a2caa&equals;0&comma;0<ac

S1:=a2aca&equals;0&comma;0<ac

(10)

out:=RealTriangularizeS1&comma;R

out:=regular_semi_algebraic_system&comma;regular_semi_algebraic_system

(11)

compl:=Complementout&comma;R

compl:=regular_semi_algebraic_system&comma;regular_semi_algebraic_system&comma;regular_semi_algebraic_system

(12)

expected:=a2caa0&comma;ac0

expected:=a2aca0&comma;ac0

(13)

expected:=mapt&rarr;opRealTriangularizet&comma;R&comma;expected

expected:=regular_semi_algebraic_system&comma;regular_semi_algebraic_system&comma;regular_semi_algebraic_system

(14)

Verify compl = expected as set of points by Difference.

Differencecompl&comma;expected&comma;R

(15)

Differenceexpected&comma;compl&comma;R

(16)

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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