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  

Let  and let  be the variety defined by F. Let  be the algebraic closure of the base field . Let phi be the projection from  to  (which ignores the first  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  under phi is a constructible set  which is the output of the command Projection(F, d, R).

If H is specified, let  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].

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.

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.