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.

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

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

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

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

$R≔\mathrm{polynomial\_ring}$

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

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

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

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

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

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

$\mathrm{dec}≔\mathrm{Triangularize}\left(F\,R\,\mathrm{output}=\mathrm{lazard}\right)$

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

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

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