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RegularChains[ChainTools]

 EquiprojectableDecomposition
 equiprojectable decomposition of a variety

 Calling Sequence EquiprojectableDecomposition(lrc, R)

Parameters

 lrc - list of regular chains of R R - polynomial ring

Description

 • The command EquiprojectableDecomposition(lrc, R) returns the equiprojectable decomposition of the variety given by lrc.
 • The variety encoded by lrc is the union of the regular zero sets of the regular chains of lrc.
 • It is assumed that every regular chain in lrc is zero-dimensional and strongly normalized.
 • This command is part of the RegularChains[ChainTools] package, so it can be used in the form EquiprojectableDecomposition(..) 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][EquiprojectableDecomposition](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[z,y,x\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $\mathrm{sys}≔\left[{x}^{2}+y+z-1,x+{y}^{2}+z-1,x+y+{z}^{2}-1\right]$
 ${\mathrm{sys}}{≔}\left[{{x}}^{{2}}{+}{y}{+}{z}{-}{1}{,}{{y}}^{{2}}{+}{x}{+}{z}{-}{1}{,}{{z}}^{{2}}{+}{x}{+}{y}{-}{1}\right]$ (2)
 > $\mathrm{lrc}≔\mathrm{Triangularize}\left(\mathrm{sys},R,\mathrm{normalized}=\mathrm{yes}\right)$
 ${\mathrm{lrc}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (3)
 > $\mathrm{map}\left(\mathrm{Equations},\mathrm{lrc},R\right)$
 $\left[\left[{z}{-}{x}{,}{y}{-}{x}{,}{{x}}^{{2}}{+}{2}{}{x}{-}{1}\right]{,}\left[{z}{,}{y}{,}{x}{-}{1}\right]{,}\left[{z}{,}{y}{-}{1}{,}{x}\right]{,}\left[{z}{-}{1}{,}{y}{,}{x}\right]\right]$ (4)
 > $\mathrm{ed}≔\mathrm{EquiprojectableDecomposition}\left(\mathrm{lrc},R\right)$
 ${\mathrm{ed}}{≔}\left\{{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right\}$ (5)
 > $\mathrm{map}\left(\mathrm{Equations},\mathrm{ed},R\right)$
 $\left\{\left[{z}{+}{y}{-}{1}{,}{{y}}^{{2}}{-}{y}{,}{x}\right]{,}\left[{2}{}{z}{+}{{x}}^{{2}}{-}{1}{,}{2}{}{y}{+}{{x}}^{{2}}{-}{1}{,}{{x}}^{{3}}{+}{{x}}^{{2}}{-}{3}{}{x}{+}{1}\right]\right\}$ (6)

References

 Dahan, X.; Moreno Maza, M.; Schost, E.; Wu, W. and Xie, Y. "Equiprojectable decompositions of zero-dimensional varieties" In proc. of International Conference on Polynomial System Solving, University of Paris 6, France, 2004.