compute the projection of a variety, a constructible set onto a specified coordinate space
compute the projection of a semi-algebraic set onto a specified coordinate space
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)
list of polynomials
list/set of equations, inequations, or inequalities
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].
First, define a polynomial ring.
R ≔ PolynomialRing⁡x,y,t
Consider the variety defined by the following two polynomials p and q.
p ≔ 5⁢t+5⁢x−y−10⁢t+7
q ≔ 5⁢t−5⁢x−t+2⁢y−7⁢t+11
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 ≔ Projection⁡p,q,1,R
To view the structure of cs, use the command RepresentingRegularSystems.
lrs ≔ RepresentingRegularSystems⁡cs,R
It consists of two components, so use the command Info to display the defining polynomials.
One component consists of a single point −1 , and the other one consists of all points except those which cancel t+1⁢t2+2⁢t+3.
Next, some examples on semi-algebraic sets will be shown.
R ≔ PolynomialRing⁡y,x
sys ≔ x2+y2−1<0
proj1 ≔ Projection⁡sys,1,R
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 ≔ RealTriangularize⁡sys,R
proj2 ≔ Projection⁡dec,1,R
The input semi-algebraic set/system can also be encoded by 4 list of polynomials.
R ≔ PolynomialRing⁡x,b,a
F ≔ x2−a⁢x+b
N ≔ x−a
H ≔ x
proj ≔ Projection⁡F,N,P,H,2,R
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][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.
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