Projection - Maple Help

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

Examples

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

First, define a polynomial ring.

 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,t\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)

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

 > $p≔\left(5t+5\right)x-y-\left(10t+7\right)$
 ${p}{≔}\left({5}{}{t}{+}{5}\right){}{x}{-}{y}{-}{10}{}{t}{-}{7}$ (2)
 > $q≔\left(5t-5\right)x-\left(t+2\right)y+\left(-7t+11\right)$
 ${q}{≔}\left({5}{}{t}{-}{5}\right){}{x}{-}\left({t}{+}{2}\right){}{y}{-}{7}{}{t}{+}{11}$ (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.

 > $\mathrm{cs}≔\mathrm{Projection}\left(\left[p,q\right],1,R\right)$
 ${\mathrm{cs}}{≔}{\mathrm{constructible_set}}$ (4)

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

 > $\mathrm{lrs}≔\mathrm{RepresentingRegularSystems}\left(\mathrm{cs},R\right)$
 ${\mathrm{lrs}}{≔}\left[{\mathrm{regular_system}}{,}{\mathrm{regular_system}}\right]$ (5)

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

 > $\mathrm{Info}\left(\mathrm{cs},R\right)$
 $\left[\left[\right]{,}\left[{t}{+}{1}{,}{{t}}^{{2}}{+}{2}{}{t}{+}{3}\right]\right]{,}\left[\left[{t}{+}{1}\right]{,}\left[{1}\right]\right]$ (6)

One component consists of a single point $-1$ , and the other one consists of all points except those which cancel $\left(t+1\right)\left({t}^{2}+2t+3\right)$.

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

 > $R≔\mathrm{PolynomialRing}\left(\left[y,x\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (7)
 > $\mathrm{sys}≔\left[{x}^{2}+{y}^{2}-1<0\right]$
 ${\mathrm{sys}}{≔}\left[{{x}}^{{2}}{+}{{y}}^{{2}}{<}{1}\right]$ (8)
 > $\mathrm{proj1}≔\mathrm{Projection}\left(\mathrm{sys},1,R\right)$
 ${\mathrm{proj1}}{≔}\left[{\mathrm{regular_semi_algebraic_system}}\right]$ (9)
 > $\mathrm{Display}\left(\mathrm{proj1},R\right)$
 $\left[{x}{<}{1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{x}{+}{1}{>}{0}\right]$ (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.

 > $\mathrm{dec}≔\mathrm{RealTriangularize}\left(\mathrm{sys},R\right)$
 ${\mathrm{dec}}{≔}\left[{\mathrm{regular_semi_algebraic_system}}\right]$ (11)
 > $\mathrm{proj2}≔\mathrm{Projection}\left(\mathrm{dec},1,R\right)$
 ${\mathrm{proj2}}{≔}\left[{\mathrm{regular_semi_algebraic_system}}\right]$ (12)
 > $\mathrm{Difference}\left(\mathrm{proj1},\mathrm{proj2},R\right)$
 $\left[\right]$ (13)
 > $\mathrm{Difference}\left(\mathrm{proj2},\mathrm{proj1},R\right)$
 $\left[\right]$ (14)

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

 > $R≔\mathrm{PolynomialRing}\left(\left[x,b,a\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (15)
 > $F≔\left[{x}^{2}-ax+b\right]$
 ${F}{≔}\left[{-}{a}{}{x}{+}{{x}}^{{2}}{+}{b}\right]$ (16)
 > $N≔\left[x-a\right]$
 ${N}{≔}\left[{x}{-}{a}\right]$ (17)
 > $P≔\left[\right]$
 ${P}{≔}\left[\right]$ (18)
 > $H≔\left[x\right]$
 ${H}{≔}\left[{x}\right]$ (19)
 > $\mathrm{proj}≔\mathrm{Projection}\left(F,N,P,H,2,R\right)$
 ${\mathrm{proj}}{≔}\left[{\mathrm{regular_semi_algebraic_system}}{,}{\mathrm{regular_semi_algebraic_system}}{,}{\mathrm{regular_semi_algebraic_system}}\right]$ (20)
 > $\mathrm{Display}\left(\mathrm{proj},R\right)$
 $\left[\left\{\begin{array}{cc}{4}{}{b}{-}{{a}}^{{2}}{=}{0}& {}\\ {a}{<}{0}& {}\end{array}\right\{,}\left\{\begin{array}{cc}{b}{=}{0}& {}\\ {a}{\ne }{0}& {}\end{array}\right\{,}\left\{\begin{array}{cc}{{a}}^{{2}}{-}{4}{}{b}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{<}{0}& {}\\ \phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{{a}}^{{2}}{-}{4}{}{b}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{\le }{0}& {}\end{array}\right\\right]$ (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.