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  $\mathrm{x1}>\mathrm{x2}>\mathrm{...}>\mathrm{xn}>\mathrm{y1}>\mathrm{...}>\mathrm{yd}$

Let $R=k\[\mathrm{x1},\mathrm{x2},...,\mathrm{xn},\mathrm{y1},...,\mathrm{yd}\]$ 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 $K^{n+d}$ to $K^d$ (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].

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$

$\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right)\:$

$\mathrm{with}\left(\mathrm{SemiAlgebraicSetTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\,t\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$p≔\left(5t\+5\right)x-y-\left(10t\+7\right)$

$p≔\left(5t+5\right)x-y-10t-7$

$q≔\left(5t-5\right)x-\left(t\+2\right)y-7t\+11$

$q≔\left(5t-5\right)x-\left(t+2\right)y-7t+11$

$\mathrm{cs}≔\mathrm{Projection}\left(\left[p\,q\right]\,1\,R\right)$

$\mathrm{cs}≔\mathrm{constructible\_set}$

$\mathrm{lrs}≔\mathrm{RepresentingRegularSystems}\left(\mathrm{cs}\,R\right)$

$\mathrm{lrs}≔\left[\mathrm{regular\_system}\,\mathrm{regular\_system}\right]$

$\mathrm{Info}\left(\mathrm{cs}\,R\right)$

$\left[\left[\right]\,\left[t+1\,t^2+2t+3\right]\right],\left[\left[t+1\right]\,\left[1\right]\right]$

$R≔\mathrm{PolynomialRing}\left(\left[y\,x\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{sys}≔\left[x^2\&plus;y^2-1<0\right]$

$\mathrm{sys}≔\left[x^2+y^2<1\right]$

$\mathrm{proj1}≔\mathrm{Projection}\left(\mathrm{sys}\&comma;1\&comma;R\right)$

$\mathrm{proj1}≔\left[\mathrm{regular\_semi\_algebraic\_system}\right]$

$\mathrm{Display}\left(\mathrm{proj1}\&comma;R\right)$

$\left[x<1\mathbf{and}x+1>0\right]$

$\mathrm{dec}≔\mathrm{RealTriangularize}\left(\mathrm{sys}\&comma;R\right)$

$\mathrm{dec}≔\left[\mathrm{regular\_semi\_algebraic\_system}\right]$

$\mathrm{proj2}≔\mathrm{Projection}\left(\mathrm{dec}\&comma;1\&comma;R\right)$

$\mathrm{proj2}≔\left[\mathrm{regular\_semi\_algebraic\_system}\right]$

$\mathrm{Difference}\left(\mathrm{proj1}\&comma;\mathrm{proj2}\&comma;R\right)$

$\mathrm{Difference}\left(\mathrm{proj2}\&comma;\mathrm{proj1}\&comma;R\right)$

$R≔\mathrm{PolynomialRing}\left(\left[x\&comma;b\&comma;a\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$F≔\left[x^2-ax\&plus;b\right]$

$F≔\left[-ax+x^2+b\right]$

$N≔\left[x-a\right]$

$N≔\left[x-a\right]$

$P≔\left[\right]$

$P≔\left[\right]$

$H≔\left[x\right]$

$H≔\left[x\right]$

$\mathrm{proj}≔\mathrm{Projection}\left(F\&comma;N\&comma;P\&comma;H\&comma;2\&comma;R\right)$

$\mathrm{proj}≔\left[\mathrm{regular\_semi\_algebraic\_system}\&comma;\mathrm{regular\_semi\_algebraic\_system}\&comma;\mathrm{regular\_semi\_algebraic\_system}\right]$

$\mathrm{Display}\left(\mathrm{proj}\&comma;R\right)$

$\left[\left\{\begin{array}{cc}4b-a^2=0& \\ a<0& \end{array}\right.\&comma;\left\{\begin{array}{cc}b=0& \\ a\ne 0& \end{array}\right.\&comma;\left\{\begin{array}{cc}{a}^2-4b>0\mathbf{and}b<0& \\ \mathbf{or}{a}^2-4b>0\mathbf{and}b>0\mathbf{and}a\le 0& \end{array}\right.\right]$

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.