The command ComprehensiveTriangularize(CS, d, R) returns a comprehensive triangular decomposition (CTD) of the constructible set CS, with respect to the last d variables of R.

The command ComprehensiveTriangularize(F, H, d, R) returns a comprehensive triangular decomposition (CTD) of the constructible set V(F)-V(H), with respect to the last d variables of R.

The command ComprehensiveTriangularize(F, d, R) returns a comprehensive triangular decomposition (CTD) of the variety V(F), with respect to the last d variables of R.

The output of ComprehensiveTriangularize(CS, d, R) consists of two parts. The first part is a pre-comprehensive triangular decomposition S of CS with respect to the last d variables of R.

The second part is a list L of pairs; the first item of each pair is a constructible set and the second item is a list of indices (positive integers) such that the union of these constructible sets forms a partition of the projection of CS onto the parameter space. Moreover, for each part (or cell) C, the list of positive integers associated with it gives the positions of the regular chains in S satisfying the following property: for each parameter value u in C, those regular chains specialized at u form a triangular decomposition of CS specialized at u.

If 'disjoint'='yes' is present, the computed CTD satisfies two additional properties. First, each regular system in S is square-free. Secondly, for each part (or cell) C, the regular chains associated with C separate disjointly at any parameter value u of C, that is, those regular chains specialize at u into square-free regular chains whose quasi-components are pairwise disjoint.

If 'output'='piecewise' is present, the output is organized as a tree. More precisely the cells in the parameter space are organized in a natural manner as a tree.

The command RealComprehensiveTriangularize(sys, d, R) returns a real comprehensive triangular decomposition (RCTD) of the semi-algebraic system sys, with respect to the last d variables of R.  These last d are regarded as the parameters of sys. The constraints in sys can be any polynomial equations, inequations or inequalities given by polynomials of R. The zero set of sys is the set of the real solutions of the polynomial system defined by sys.

The command RealComprehensiveTriangularize(F, N, P, H, d, R) returns a RCTD of the semi-algebraic system defined by F, N, P, H and with respect to the last d variables of R.  See SemiAlgebraicSetTools or RealTriangularize for the specification of the semi-algebraic system defined by four polynomial lists F, N, P, H.

All uses of RealComprehensiveTriangularize assume that R has characteristic zero and no parameters, such that the base field of R is the field of rational numbers.

The output of RealComprehensiveTriangularize(sys, d, R) or RealComprehensiveTriangularize(F, N, P, H, d, R) is a list of two items.

The first one is a list systems of square-free triangular semi-algebraic systems and the second one is a list cells of semi-algebraic sets forming a partition of the parameter space. Above each of these semi-algebraic sets, the solutions of the input semi-algebraic system is described by a subset of systems, using the same indexing convention as for ComprehensiveTriangularize.

Each of these semi-algebraic sets is a disjoint union of CAD cells; these cells are connected such that the solutions above a cell depend continuously on the parameters of that cell. Moreover, if the input system admits finitely many solutions above that cell, then the graphs of the functions (from the corresponding items in systems) describing those solutions are disjoint.

These properties make it possible to count the number of real solutions of the input system depending on the parameters. For this reason the output rctd of RealComprehensiveTriangularize(sys, d, R) or RealComprehensiveTriangularize(F, N, P, H, d, R) can be used as first input argument of RealComprehensiveTriangularize together with a prescribed number or number range of solution. In this case only the items of systems and cells corresponding to the prescribed number or number range are returned. See the examples below.

output=cadcell and output=semialgebraicset options controls the representation of the output format.  The default is semialgebraicset.

For a mathematical definition of the type regular_chain, see the page RegularChains.

For mathematical definitions of the types regular_system and constructible_set, see the page ConstructibleSetTools.

For mathematical definitions of the types squarefree_semi_algebraic_system, semi_algebraic_set and cad_cell, see the page SemiAlgebraicSetTools.

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

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

$R≔\mathrm{polynomial\_ring}$

$F≔\left[s-\left(y+1\right)x\,s-\left(x+1\right)y\right]$

$F≔\left[s-\left(y+1\right)x\,s-\left(x+1\right)y\right]$

$\mathrm{pctd},\mathrm{cells}≔\mathrm{ComprehensiveTriangularize}\left(F\,1\,R\right)$

$\mathrm{pctd},\mathrm{cells}≔\left[\mathrm{regular\_chain}\,\mathrm{regular\_chain}\,\mathrm{regular\_chain}\right],\left[\left[\mathrm{constructible\_set}\,\left[3\,2\right]\right]\,\left[\mathrm{constructible\_set}\,\left[1\right]\right]\right]$

$\mathrm{pctd}$

$\left[\mathrm{regular\_chain}\,\mathrm{regular\_chain}\,\mathrm{regular\_chain}\right]$

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

$\left[\left[\left(y+1\right)x-s\,y^2+y-s\right]\,\left[x+1\,y+1\,s\right]\,\left[x\,y\,s\right]\right]$

$\mathrm{cs1}≔\mathrm{cells}\left[1\right]\left[1\right]\;$$\mathrm{Info}\left(\mathrm{cs1}\,R\right)$

$\mathrm{cs1}≔\mathrm{constructible\_set}$

$\left[\left[s\right]\,\left[1\right]\right]$

$\mathrm{cs2}≔\mathrm{cells}\left[2\right]\left[1\right]\;$$\mathrm{Info}\left(\mathrm{cs2}\,R\right)$

$\mathrm{cs2}≔\mathrm{constructible\_set}$

$\left[\left[\right]\,\left[s\right]\right]$

$F≔\left[s-\left(y+1\right)x\,s-\left(x+1\right)y\right]$

$F≔\left[s-\left(y+1\right)x\,s-\left(x+1\right)y\right]$

$H≔\left[x+y-1\right]$

$H≔\left[x+y-1\right]$

$\mathrm{cs}≔\mathrm{GeneralConstruct}\left(F\,H\,R\right)$

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

$\mathrm{ComprehensiveTriangularize}\left(\mathrm{cs}\,1\,R\right)$

$\left[\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\right],\left[\left[\mathrm{constructible\_set}\,\left[1\right]\right]\,\left[\mathrm{constructible\_set}\,\left[2\,4\right]\right]\,\left[\mathrm{constructible\_set}\,\left[3\right]\right]\right]$

$\mathrm{ComprehensiveTriangularize}\left(\mathrm{cs}\,1\,R\,\mathrm{disjoint}=\mathrm{yes}\right)$

$\left[\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\,\mathrm{regular\_system}\right],\left[\left[\mathrm{constructible\_set}\,\left[1\right]\right]\,\left[\mathrm{constructible\_set}\,\left[2\,4\right]\right]\,\left[\mathrm{constructible\_set}\,\left[5\right]\right]\,\left[\mathrm{constructible\_set}\,\left[3\right]\right]\right]$

$\mathrm{ComprehensiveTriangularize}\left(F\,H\,1\,R\,\mathrm{output}=\mathrm{piecewise}\right)$

$\left\{\begin{array}{cc}\left[\left\{y^2+y-s=0\,xy-s+x=0\,y+1\ne 0\,-y+2s-1\ne 0\right\}\right]& s\ne 0\wedge 4s-3\ne 0\\ \left[\left\{s=0\,x+1=0\,y+1=0\right\}\,\left\{s=0\,x=0\,y=0\right\}\right]& s=0\\ \left[\left\{4s-3=0\,2x+3=0\,2y+3=0\right\}\right]& 4s-3=0\end{array}\right.$

$\mathrm{ComprehensiveTriangularize}\left(F\,H\,1\,R\,\mathrm{disjoint}=\mathrm{yes}\,\mathrm{output}=\mathrm{piecewise}\right)$

$\left\{\begin{array}{cc}\left[\left\{y^2+y-s=0\,xy-s+x=0\,y+1\ne 0\,-y+2s-1\ne 0\right\}\right]& s\ne 0\wedge 4s-3\ne 0\wedge 4s+1\ne 0\\ \left[\left\{s=0\,x+1=0\,y+1=0\right\}\,\left\{s=0\,x=0\,y=0\right\}\right]& s=0\\ \left[\left\{4s-3=0\,2x+3=0\,2y+3=0\right\}\right]& 4s-3=0\\ \left[\left\{4s+1=0\,2x+1=0\,2y+1=0\right\}\right]& 4s+1=0\end{array}\right.$

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

$R≔\mathrm{polynomial\_ring}$

$f≔a{x}^2+bx+c$

$f≔a{x}^2+bx+c$

$\mathrm{rctd}≔\mathrm{RealComprehensiveTriangularize}\left(\left[f=0\&comma;0<a\right]\&comma;3\&comma;R\right)$

$\mathrm{rctd}≔\left[\left[\left[1\&comma;\mathrm{squarefree\_semi\_algebraic\_system}\right]\&comma;\left[2\&comma;\mathrm{squarefree\_semi\_algebraic\_system}\right]\right]\&comma;\left[\left[\mathrm{semi\_algebraic\_set}\&comma;\left[\right]\right]\&comma;\left[\mathrm{semi\_algebraic\_set}\&comma;\left[1\right]\right]\&comma;\left[\mathrm{semi\_algebraic\_set}\&comma;\left[2\right]\right]\right]\right]$

$\mathrm{Display}\left(\mathrm{rctd}\left[1\right]\&comma;R\right)$

$\left[\left[1\&comma;a{x}^2+bx+c=0\right]\&comma;\left[2\&comma;2ax+b=0\right]\right]$

$\mathrm{Display}\left(\mathrm{rctd}\left[2\right]\&comma;R\right)$

$\left[\left[\left[\left\{\begin{array}{cc}c<\frac{b^2}{4a}& \\ b=b& \\ a<0& \end{array}\right.\&comma;\left\{\begin{array}{cc}c=\frac{b^2}{4a}& \\ b=b& \\ a<0& \end{array}\right.\&comma;\left\{\begin{array}{cc}\frac{b^2}{4a}<c& \\ b=b& \\ a<0& \end{array}\right.\&comma;\left\{\begin{array}{cc}c=c& \\ b=b& \\ a=0& \end{array}\right.\&comma;\left\{\begin{array}{cc}\frac{b^2}{4a}<c& \\ b=b& \\ 0<a& \end{array}\right.\right]\&comma;\left[\right]\right]\&comma;\left[\left[\left\{\begin{array}{cc}c<\frac{b^2}{4a}& \\ b=b& \\ 0<a& \end{array}\right.\right]\&comma;\left[1\right]\right]\&comma;\left[\left[\left\{\begin{array}{cc}c=\frac{b^2}{4a}& \\ b=b& \\ 0<a& \end{array}\right.\right]\&comma;\left[2\right]\right]\right]$

$\mathrm{rctd}≔\mathrm{RealComprehensiveTriangularize}\left(\mathrm{rctd}\&comma;R\&comma;2\right)$

$\mathrm{rctd}≔\left[\left[\left[1\&comma;\mathrm{squarefree\_semi\_algebraic\_system}\right]\right]\&comma;\left[\left[\mathrm{semi\_algebraic\_set}\&comma;\left[1\right]\right]\right]\right]$

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

$\left[\left[\left[1\&comma;a{x}^2+bx+c=0\right]\right]\&comma;\left[\left[\left[\left\{\begin{array}{cc}c<\frac{b^2}{4a}& \\ b=b& \\ 0<a& \end{array}\right.\right]\&comma;\left[1\right]\right]\right]\right]$

$\mathrm{rctd}≔\mathrm{RealComprehensiveTriangularize}\left(\left[f=0\right]\&comma;3\&comma;R\&comma;\mathrm{\infty }\right)$

$\mathrm{rctd}≔\left[\left[\left[1\&comma;\mathrm{squarefree\_semi\_algebraic\_system}\right]\right]\&comma;\left[\left[\mathrm{semi\_algebraic\_set}\&comma;\left[1\right]\right]\right]\right]$

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

$\left[\left[\left[1\&comma;\mathrm{true}\right]\right]\&comma;\left[\left[\left[\left\{\begin{array}{cc}c=0& \\ b=0& \\ a=0& \end{array}\right.\right]\&comma;\left[1\right]\right]\right]\right]$

$\mathrm{rctd}≔\mathrm{RealComprehensiveTriangularize}\left(\left[f=0\right]\&comma;3\&comma;R\&comma;2\right)$

$\mathrm{rctd}≔\left[\left[\left[1\&comma;\mathrm{squarefree\_semi\_algebraic\_system}\right]\right]\&comma;\left[\left[\mathrm{semi\_algebraic\_set}\&comma;\left[1\right]\right]\right]\right]$

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

$\left[\left[\left[1\&comma;a{x}^2+bx+c=0\right]\right]\&comma;\left[\left[\left[\left\{\begin{array}{cc}\frac{b^2}{4a}<c& \\ b=b& \\ a<0& \end{array}\right.\&comma;\left\{\begin{array}{cc}c<\frac{b^2}{4a}& \\ b=b& \\ 0<a& \end{array}\right.\right]\&comma;\left[1\right]\right]\right]\right]$

$\mathrm{rctd}≔\mathrm{RealComprehensiveTriangularize}\left(\left[f=0\right]\&comma;3\&comma;R\&comma;2\&comma;\mathrm{output}='\mathrm{cadcell}'\right)$

$\mathrm{rctd}≔\left[\left[\left[1\&comma;\mathrm{squarefree\_semi\_algebraic\_system}\right]\right]\&comma;\left[\left[\mathrm{cad\_cell}\&comma;\left[1\right]\right]\&comma;\left[\mathrm{cad\_cell}\&comma;\left[1\right]\right]\right]\right]$

$R≔\mathrm{PolynomialRing}\left(\left[x\&comma;c\&comma;b\&comma;a\right]\right)\&semi;$$f≔a{x}^3+bx+c$

$R≔\mathrm{polynomial\_ring}$

$f≔a{x}^3+xb+c$

$\mathrm{rctd}≔\mathrm{RealComprehensiveTriangularize}\left(\left[f=0\&comma;0<a\right]\&comma;3\&comma;R\right)$

$\mathrm{rctd}≔\left[\left[\left[1\&comma;\mathrm{squarefree\_semi\_algebraic\_system}\right]\&comma;\left[2\&comma;\mathrm{squarefree\_semi\_algebraic\_system}\right]\&comma;\left[3\&comma;\mathrm{squarefree\_semi\_algebraic\_system}\right]\right]\&comma;\left[\left[\mathrm{semi\_algebraic\_set}\&comma;\left[\right]\right]\&comma;\left[\mathrm{semi\_algebraic\_set}\&comma;\left[1\right]\right]\&comma;\left[\mathrm{semi\_algebraic\_set}\&comma;\left[2\right]\right]\&comma;\left[\mathrm{semi\_algebraic\_set}\&comma;\left[3\right]\right]\right]\right]$

$\mathrm{Display}\left(\mathrm{rctd}\left[1\right]\&comma;R\right)$

$\left[\left[1\&comma;a{x}^3+bx+c=0\right]\&comma;\left[2\&comma;4b^{2}a{x}^2-6cbax+9c^{2}a+4b^3=0\right]\&comma;\left[3\&comma;x=0\right]\right]$

$\mathrm{Display}\left(\mathrm{rctd}\left[2\right]\&comma;R\right)$

$\left[\left[\left[\left\{\begin{array}{cc}c=c& \\ b<0& \\ a<0& \end{array}\right.\&comma;\left\{\begin{array}{cc}c<0& \\ b=0& \\ a<0& \end{array}\right.\&comma;\left\{\begin{array}{cc}c=0& \\ b=0& \\ a<0& \end{array}\right.\&comma;\left\{\begin{array}{cc}0<c& \\ b=0& \\ a<0& \end{array}\right.\&comma;\left\{\begin{array}{cc}c<\frac{2\sqrt{-3a{b}^3}}{9a}& \\ 0<b& \\ a<0& \end{array}\right.\&comma;\left\{\begin{array}{cc}c=\frac{2\sqrt{-3a{b}^3}}{9a}& \\ 0<b& \\ a<0& \end{array}\right.\&comma;\left\{\begin{array}{cc}\frac{2\sqrt{-3a{b}^3}}{9a}<c\wedge c<-\frac{2\sqrt{-3a{b}^3}}{9a}& \\ 0<b& \\ a<0& \end{array}\right.\&comma;\left\{\begin{array}{cc}c=-\frac{2\sqrt{-3a{b}^3}}{9a}& \\ 0<b& \\ a<0& \end{array}\right.\&comma;\left\{\begin{array}{cc}-\frac{2\sqrt{-3a{b}^3}}{9a}<c& \\ 0<b& \\ a<0& \end{array}\right.\&comma;\left\{\begin{array}{cc}c=c& \\ b=b& \\ a=0& \end{array}\right.\right]\&comma;\left[\right]\right]\&comma;\left[\left[\left\{\begin{array}{cc}c<-\frac{2\sqrt{-3a{b}^3}}{9a}& \\ b<0& \\ 0<a& \end{array}\right.\&comma;\left\{\begin{array}{cc}-\frac{2\sqrt{-3a{b}^3}}{9a}<c\wedge c<\frac{2\sqrt{-3a{b}^3}}{9a}& \\ b<0& \\ 0<a& \end{array}\right.\&comma;\left\{\begin{array}{cc}\frac{2\sqrt{-3a{b}^3}}{9a}<c& \\ b<0& \\ 0<a& \end{array}\right.\&comma;\left\{\begin{array}{cc}c<0& \\ b=0& \\ 0<a& \end{array}\right.\&comma;\left\{\begin{array}{cc}0<c& \\ b=0& \\ 0<a& \end{array}\right.\&comma;\left\{\begin{array}{cc}c=c& \\ 0<b& \\ 0<a& \end{array}\right.\right]\&comma;\left[1\right]\right]\&comma;\left[\left[\left\{\begin{array}{cc}c=-\frac{2\sqrt{-3a{b}^3}}{9a}& \\ b<0& \\ 0<a& \end{array}\right.\&comma;\left\{\begin{array}{cc}c=\frac{2\sqrt{-3a{b}^3}}{9a}& \\ b<0& \\ 0<a& \end{array}\right.\right]\&comma;\left[2\right]\right]\&comma;\left[\left[\left\{\begin{array}{cc}c=0& \\ b=0& \\ 0<a& \end{array}\right.\right]\&comma;\left[3\right]\right]\right]$

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.; and Moreno Maza, M. "Semi-algebraic description of the equilibria of dynamical systems." Proc. Cof the 2011 International Workshop on Computer Algebra in Scientific Computing  (CASC 2011),  LNCS Vol. 6885: 101-125. Springer, 2011.

The RegularChains[ParametricSystemTools][RealComprehensiveTriangularize] command was introduced in Maple 16.

The sys, N, P, rctd, n, 'infinity', 'finite', 'output'= and 'output'='cadcell' parameters were introduced in Maple 16.

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