 PreComprehensiveTriangularize - Maple Help

RegularChains[ParametricSystemTools]

 PreComprehensiveTriangularize
 compute a pre-comprehensive triangular decomposition Calling Sequence PreComprehensiveTriangularize(sys, d, R) Parameters

 sys - list of polynomials d - number of parameters R - polynomial ring Description

 • The command PreComprehensiveTriangularize(sys, d, R)  returns a pre-comprehensive triangular decomposition of sys, with respect to the last d variables of R.
 • A pre-comprehensive triangular decomposition is a refined triangular decomposition (in the Lazard sense) with additional properties, aiming at studying parametric polynomial systems.
 • Let $U$ be the last d variables of R, which we regard as parameters. A finite set $S$ of regular chains of R forms a pre-comprehensive triangular decomposition of F with respect to U, if for every parameter value $u$, there exists a subset $S\left(u\right)$ of $S$ such that
 (1) the regular chains of $S\left(u\right)$ specialize well at $u$, and
 (2) after specialization at $u$, these chains form a triangular decomposition (in the Lazard sense) of the polynomial system $F$ specialized at $u$. See the command DefiningSet for the term specialize well. Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right):$
 > $\mathrm{with}\left(\mathrm{ParametricSystemTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,s\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $F≔\left[s-\left(y+1\right)x,s-\left(x+1\right)y\right]$
 ${F}{≔}\left[{s}{-}\left({y}{+}{1}\right){}{x}{,}{s}{-}\left({x}{+}{1}\right){}{y}\right]$ (2)

A pre-comprehensive triangular decomposition of $F$ consists of three regular chains.

 > $\mathrm{pctd}≔\mathrm{PreComprehensiveTriangularize}\left(F,1,R\right)$
 ${\mathrm{pctd}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (3)
 > $\mathrm{map}\left(\mathrm{Info},\mathrm{pctd},R\right)$
 $\left[\left[\left({y}{+}{1}\right){}{x}{-}{s}{,}{{y}}^{{2}}{+}{y}{-}{s}\right]{,}\left[{x}{+}{1}{,}{y}{+}{1}{,}{s}\right]{,}\left[{x}{,}{y}{,}{s}\right]\right]$ (4)

Compare it with the output of Triangularize.

 > $\mathrm{dec}≔\mathrm{Triangularize}\left(F,R,\mathrm{output}=\mathrm{lazard}\right)$
 ${\mathrm{dec}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (5)
 > $\mathrm{map}\left(\mathrm{Info},\mathrm{dec},R\right)$
 $\left[\left[\left({y}{+}{1}\right){}{x}{-}{s}{,}{{y}}^{{2}}{+}{y}{-}{s}\right]{,}\left[{x}{+}{1}{,}{y}{+}{1}{,}{s}\right]\right]$ (6)