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RegularChains[ChainTools]

 SeparateSolutions
 decomposition into pairwise disjoint regular chains

 Calling Sequence SeparateSolutions(l_rc, R)

Parameters

 l_rc - list of regular chains R - polynomial ring

Description

 • The command SeparateSolutions(l_rc, R) returns a list of square-free regular chains such that the ideals they generate are pairwise relatively prime.
 • The input regular chains must be zero-dimensional.
 • The algorithm is based on GCD computations.
 • This command is part of the RegularChains[ChainTools] package, so it can be used in the form SeparateSolutions(..) only after executing the command with(RegularChains[ChainTools]). However, it can always be accessed through the long form of the command by using RegularChains[ChainTools][SeparateSolutions](..).

Examples

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

Consider a polynomial ring with two variables

 > $R≔\mathrm{PolynomialRing}\left(\left[y,x\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)

Consider two regular chains in R

 > $\mathrm{rc1}≔\mathrm{Chain}\left(\left[x,y\left(y+1\right)\right],\mathrm{Empty}\left(R\right),R\right)$
 ${\mathrm{rc1}}{≔}{\mathrm{regular_chain}}$ (2)
 > $\mathrm{rc2}≔\mathrm{Chain}\left(\left[x,y\left(y+2\right)\right],\mathrm{Empty}\left(R\right),R\right)$
 ${\mathrm{rc2}}{≔}{\mathrm{regular_chain}}$ (3)

These two regular chains share a common solution. The union of their zero sets can be made disjoint. In other words we can replace these two regular chains by another set of regular chains such that the two sets describe the same solutions and the second one consists of pairwise disjoint zero sets of regular chains. This is done as follows

 > $\mathrm{out}≔\mathrm{SeparateSolutions}\left(\left[\mathrm{rc1},\mathrm{rc2}\right],R\right)$
 ${\mathrm{out}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (4)
 > $\mathrm{map}\left(\mathrm{Equations},\mathrm{out},R\right)$
 $\left[\left[{y}{+}{1}{,}{x}\right]{,}\left[{y}{,}{x}\right]{,}\left[{y}{+}{2}{,}{x}\right]\right]$ (5)

 See Also