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RegularChains[ConstructibleSetTools][Projection] - compute the projection of a variety, a constructible set onto a specified coordinate space

RegularChains[SemiAlgebraicSetTools][Projection] - compute the projection of a semi-algebraic set onto a specified coordinate space

Calling Sequence

Projection(F, d, R)

Projection(F, H, d, R)

Projection(CS, d, R)

Projection(sys, d, R)

Projection(lrsas, d, R)

Projection(F,N,P,H, d, R)

Parameters

F

-

list of polynomials

d

-

positive integer

R

-

polynomial ring

H

-

list of polynomials

CS

-

constructible set

sys

-

list/set of equations, inequations, or inequalities

lrsas

-

list/set of equations, inequations, or inequalities

N

-

list of polynomials

P

-

list of polynomials

Description

• 

The subcoordinate space is specified by the parameters d and R. The parameters d must be less than the number of variables and d must be at least 1. For an algebraic variety or a constructible sets, the ring may have characteristic zero or a prime characteristic; for semi-algebraic sets, the ring must have characteristic zero.

• 

The projection can be applied to either a constructible set (or an algebraic variety), or a semi-algebraic set (encoded by a list of regular_semi_algebraic_system or four list of polynomials). The projection image of a constructible set is an constructible set, encoded as a constructible_set object; the projection image of a semi-algebraic set is a semi-algebraic set, encoded as a list of regular_semi_algebraic_system. The variables in R are ordered as  x1>x2>...>xn>y1>...>yd

• 

Let R=k[x1,x2,...,xn,y1,...,yd] and let V be the variety defined by F. Let K be the algebraic closure of the base field k. Let phi be the projection from Kn+d to Kd (which ignores the first n coordinates).

  

Then the command Projection(F, d, R) returns the image of the variety defined by F under the d-th standard projection. The image of V under phi is a constructible set C which is the output of the command Projection(F, d, R).

• 

If H is specified, let W be the variety defined by the product of polynomials in H.  Then the command Projection(F, H, d, R) returns the image of the constructible set defined by the difference of V and W under the d-th standard projection.

• 

The command Projection(CS, d, R)  returns the image of the constructible set CS under the d-th standard projection.

• 

The command Projection(F, N, P, H, d, R)  returns the image of the zero set of the semi-algebraic system encoded by [F,N,P,H], see SemiAlgebraicSetTools or  RealTriangularize.

• 

The command Projection(sys, d, R)  returns the image of the semi-algebraic set defined by the constraints in sys.

• 

The command Projection(lrsas, d, R)  returns the image of the semi-algebraic union of zeros sets of the regular semi-algebraic systems in lrsas.

• 

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][Projection] or RegularChains[SemiAlgebraicSetTools][Projection].

Examples

withRegularChains:

withConstructibleSetTools:

withSemiAlgebraicSetTools:

First, define a polynomial ring.

R:=PolynomialRingx,y,t

R:=polynomial_ring

(1)

Consider the variety defined by the following two polynomials p and q.

p:=5t+5xy10t+7

p:=5t+5xy10t7

(2)

q:=5t5xt+2y7t+11

q:=5t5xt+2y7t+11

(3)

Now set d=1, meaning that the projection is to the coordinate space of t. The projection of V to K is given by the following constructible set cs.

cs:=Projectionp,q,1,R

cs:=constructible_set

(4)

To view the structure of cs, use the command RepresentingRegularSystems.

lrs:=RepresentingRegularSystemscs,R

lrs:=regular_system,regular_system

(5)

It consists of two components, so use the command Info to display the defining polynomials.

Infocs,R

,t+1,t2+2t+3,t+1,1

(6)

One component consists of a single point 1 , and the other one consists of all points except those which cancel t+1t2+2t+3.

Next, some examples on semi-algebraic sets will be shown.

R:=PolynomialRingy,x

R:=polynomial_ring

(7)

sys:=x2&plus;y21<0

sys:=x2&plus;y2<1

(8)

proj1:=Projectionsys&comma;1&comma;R

proj1:=regular_semi_algebraic_system

(9)

Displayproj1&comma;R

x<1andx+1>0

(10)

One can always turn a input semi-algebraic system to a list of regular semi-algebraic system (called a triangular decomposition) by RealTriangularize, and the compute the Projection.

dec:=RealTriangularizesys&comma;R

dec:=regular_semi_algebraic_system

(11)

proj2:=Projectiondec&comma;1&comma;R

proj2:=regular_semi_algebraic_system

(12)

Differenceproj1&comma;proj2&comma;R

(13)

Differenceproj2&comma;proj1&comma;R

(14)

The input semi-algebraic set/system can also be encoded by 4 list of polynomials.

R:=PolynomialRingx&comma;b&comma;a

R:=polynomial_ring

(15)

F:=x2ax&plus;b

F:=ax&plus;x2&plus;b

(16)

N:=xa

N:=xa

(17)

P:=

P:=

(18)

H:=x

H:=x

(19)

proj:=ProjectionF&comma;N&comma;P&comma;H&comma;2&comma;R

proj:=regular_semi_algebraic_system&comma;regular_semi_algebraic_system&comma;regular_semi_algebraic_system

(20)

Displayproj&comma;R

&lcub;4ba2=0a<0&comma;&lcub;b=0a0&comma;&lcub;a24b>0andb<0ora24b>0andb>0anda0

(21)

See Also

Complement, ConstructibleSetTools, Difference, Intersection, RealTriangularize, 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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