The command RefiningPartition(lcs, R) returns a list of pairwise disjoint constructible sets out_lcs such that every constructible set of lcs can be written as a disjoint union of several constructible sets in out_lcs.

This is represented by a matrix showing the constructible set and its associated indices where the constructible set comes from.

This function can also be seen as a set theoretical instance of the co-prime factorization problem.

This command is part of the RegularChains[ConstructibleSetTools] package, so it can be used in the form RefiningPartition(..) only after executing the command with(RegularChains[ConstructibleSetTools]). However, it can always be accessed through the long form of the command by using RegularChains[ConstructibleSetTools][RefiningPartition](..).

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

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

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

$R≔\mathrm{polynomial\_ring}$

$F≔\left[vxy+u{x}^2+x\,u{y}^2+x^2\right]$

$F≔\left[u{x}^2+vxy+x\,u{y}^2+x^2\right]$

$\mathrm{cs1}≔\mathrm{GeneralConstruct}\left(F\,\left[u\right]\,R\right)$

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

$\mathrm{cs2}≔\mathrm{GeneralConstruct}\left(F\,\left[v\right]\,R\right)$

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

$\mathrm{cs3}≔\mathrm{Projection}\left(F\,2\,R\right)$

$\mathrm{cs3}≔\mathrm{constructible\_set}$

$\mathrm{RefiningPartition}\left(\left[\mathrm{cs1}\,\mathrm{cs2}\right]\,R\right)$

$\left[\begin{array}{cc}\mathrm{constructible\_set}& \left[1\right]\\ \mathrm{constructible\_set}& \left[1\,2\right]\\ \mathrm{constructible\_set}& \left[2\right]\end{array}\right]$

$\mathrm{RefiningPartition}\left(\left[\mathrm{cs1}\,\mathrm{cs2}\,\mathrm{cs3}\right]\,R\right)$

$\left[\begin{array}{cc}\mathrm{constructible\_set}& \left[3\right]\\ \mathrm{constructible\_set}& \left[3\,1\right]\\ \mathrm{constructible\_set}& \left[3\,1\,2\right]\\ \mathrm{constructible\_set}& \left[3\,2\right]\end{array}\right]$

Chen, C.; Golubitsky, O.; Lemaire, F.; Moreno Maza, M.; and Pan, W. "Comprehensive Triangular Decomposition." Proc. CASC 2007, LNCS, Vol. 4770, pp. 73-101. Springer, 2007.