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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.

Examples

withRegularChains:

withConstructibleSetTools:

withSemiAlgebraicSetTools:

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

R:=PolynomialRingx,y,t

R:=polynomial_ring

(1)

p:=5t+5xy10t+7

p:=5t+5xy10t7

(2)

q:=5t5xt+2y7t+11

q:=5t5xt+2y7t+11

(3)

Using the GeneralConstruct command and adding one inequality, you can build a constructible set. Using the polynomials xt and x+t for defining inequations, the two constructible sets cs1 and cs2 are different.

cs1:=GeneralConstructp,q,xt,R

cs1:=constructible_set

(4)

cs2:=GeneralConstructp,q,x+t,R

cs2:=constructible_set

(5)

The intersection of cs1 and cs2 is a new constructible set cs.

cs:=Intersectioncs1,cs2,R

cs:=constructible_set

(6)

Check the result in another way.

cs3:=GeneralConstructp,q,x+t,xt,R

cs3:=constructible_set

(7)

IsContainedcs3,cs,RandIsContainedcs,cs3,R

true

(8)

The results are as desired.

Consider now the semi-algebraic case:

lrsas1:=RealTriangularizep,q,,,xt,R

lrsas1:=regular_semi_algebraic_system,regular_semi_algebraic_system

(9)

lrsas2:=RealTriangularizep,q,,,x+t,R

lrsas2:=regular_semi_algebraic_system,regular_semi_algebraic_system

(10)

lrsas:=Intersectionlrsas1,lrsas2,R

lrsas:=regular_semi_algebraic_system,regular_semi_algebraic_system

(11)

lrsas12:=RealTriangularizep,q,,,x+t,xt,R

lrsas12:=regular_semi_algebraic_system,regular_semi_algebraic_system

(12)

Verify the results

Differencelrsas,lrsas12,R

(13)

Differencelrsas12,lrsas,R

(14)

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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