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Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : SemiAlgebraicSetTools Subpackage : RegularChains/ConstructibleSetTools/Projection

RegularChains

  

ConstructibleSetTools[Projection]

  

compute the projection of a variety, a constructible set onto a specified coordinate space

  

SemiAlgebraicSetTools[Projection]

  

compute the projection of a semi-algebraic set onto a specified coordinate space

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

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.

RPolynomialRingx,y,t

R:=polynomial_ring

(1)

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

p5t+5xy10t+7

p:=5t+5xy10t7

(2)

q5t5xt+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.

csProjectionp,q,1,R

cs:=constructible_set

(4)

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

lrsRepresentingRegularSystemscs,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.

RPolynomialRingy,x

R:=polynomial_ring

(7)

sysx2&plus;y21<0

sys:=x2&plus;y2<1

(8)

proj1Projectionsys&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.

decRealTriangularizesys&comma;R

dec:=regular_semi_algebraic_system

(11)

proj2Projectiondec&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.

RPolynomialRingx&comma;b&comma;a

R:=polynomial_ring

(15)

Fx2ax&plus;b

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

(16)

Nxa

N:=xa

(17)

P

P:=

(18)

Hx

H:=x

(19)

projProjectionF&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)

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.

Compatibility

• 

The RegularChains[SemiAlgebraicSetTools][Projection] command was introduced in Maple 16.

• 

The sys, lrsas, N and P parameters were introduced in Maple 16.

• 

For more information on Maple 16 changes, see Updates in Maple 16.

See Also

Complement

ConstructibleSetTools

Difference

Intersection

RealTriangularize

RegularChains

SemiAlgebraicSetTools

 


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