compute the coprime factorization of a list of constructible sets - Maple Help

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RegularChains[ConstructibleSetTools][RefiningPartition] - compute the coprime factorization of a list of constructible sets

 Calling Sequence RefiningPartition(lcs, R)

Parameters

 lcs - list of constructible sets R - polynomial ring

Description

 • 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](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right):$
 > $R:=\mathrm{PolynomialRing}\left(\left[x,y,u,v\right]\right)$
 ${R}{:=}{\mathrm{polynomial_ring}}$ (1)
 > $F:=\left[vxy+u{x}^{2}+x,u{y}^{2}+{x}^{2}\right]$
 ${F}{:=}\left[{u}{}{{x}}^{{2}}{+}{v}{}{x}{}{y}{+}{x}{,}{u}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right]$ (2)
 > $\mathrm{cs1}:=\mathrm{GeneralConstruct}\left(F,\left[u\right],R\right)$
 ${\mathrm{cs1}}{:=}{\mathrm{constructible_set}}$ (3)
 > $\mathrm{cs2}:=\mathrm{GeneralConstruct}\left(F,\left[v\right],R\right)$
 ${\mathrm{cs2}}{:=}{\mathrm{constructible_set}}$ (4)
 > $\mathrm{cs3}:=\mathrm{Projection}\left(F,2,R\right)$
 ${\mathrm{cs3}}{:=}{\mathrm{constructible_set}}$ (5)
 > $\mathrm{RefiningPartition}\left(\left[\mathrm{cs1},\mathrm{cs2}\right],R\right)$
 $\left[\begin{array}{cc}{\mathrm{constructible_set}}& \left[{1}\right]\\ {\mathrm{constructible_set}}& \left[{1}{,}{2}\right]\\ {\mathrm{constructible_set}}& \left[{2}\right]\end{array}\right]$ (6)
 > $\mathrm{RefiningPartition}\left(\left[\mathrm{cs1},\mathrm{cs2},\mathrm{cs3}\right],R\right)$
 $\left[\begin{array}{cc}{\mathrm{constructible_set}}& \left[{3}\right]\\ {\mathrm{constructible_set}}& \left[{3}{,}{1}\right]\\ {\mathrm{constructible_set}}& \left[{3}{,}{1}{,}{2}\right]\\ {\mathrm{constructible_set}}& \left[{3}{,}{2}\right]\end{array}\right]$ (7)