RegularChains - Maple Programming Help

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RegularChains

 ChainTools[RemoveRedundantComponents]
 remove redundant quasi-components from a list of regular chains
 SemiAlgebraicSetTools[RemoveRedundantComponents]
 remove redundant quasi-components from a list of regular semi-algebraic systems

 Calling Sequence RemoveRedundantComponents(lrc, R) RemoveRedundantComponents(lrsas, R)

Parameters

 lrc - list of regular chains lrsas - list of regular semi-algebraic systems R - polynomial ring

Description

 • The command RemoveRedundantComponents(lrc, R) returns a list $\mathrm{lrc2}$ of regular chains whose quasi-components are pairwise noninclusive and such that lrc and $\mathrm{lrc2}$ are Lazard decompositions of the same algebraic variety. Consequently, this command removes from $\mathrm{lrc2}$ those quasi-components that are redundant for inclusion.
 • The command RemoveRedundantComponents(lrsas, R) returns a list $\mathrm{res}$ of regular semi-algebraic system whose zero sets are pairwise noninclusive, and such that lrsas and $\mathrm{res}$ have the same zero set.
 • For more details, see Algorithm 35 in the Ph.D. thesis of Yuzhen Xie.

Examples

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

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

The solutions of one are contained in those of the other. The redundant one will be removed as follows

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

The case of semi-algebraic system.

 > $\mathrm{C1}≔\left[0
 > $\mathrm{C2}≔\left[0
 > $\mathrm{C3}≔\left[a-c<0,0
 > $S≔\left[\mathrm{C1},\mathrm{C2},\mathrm{C3}\right]$
 ${S}{≔}\left[\left[{0}{<}{a}{,}{0}{<}{b}{,}{0}{<}{c}{,}{a}{<}{b}{+}{c}{,}{b}{<}{a}{+}{c}{,}{c}{<}{a}{+}{b}{,}{{a}}^{{2}}{+}{{b}}^{{2}}{-}{{c}}^{{2}}{\le }{0}\right]{,}\left[{0}{<}{a}{,}{0}{<}{b}{,}{0}{<}{c}{,}{a}{<}{b}{+}{c}{,}{b}{<}{a}{+}{c}{,}{c}{<}{a}{+}{b}{,}{c}{}{\left({{a}}^{{2}}{+}{{b}}^{{2}}{-}{{c}}^{{2}}\right)}^{{2}}{<}{a}{}{{b}}^{{2}}{}\left({-}{{a}}^{{2}}{+}{2}{}{a}{}{c}{+}{{b}}^{{2}}{-}{{c}}^{{2}}\right)\right]{,}\left[{a}{-}{c}{<}{0}{,}{0}{<}{a}{,}{0}{<}{b}{,}{0}{<}{c}{,}{a}{<}{b}{+}{c}{,}{b}{<}{a}{+}{c}{,}{c}{<}{a}{+}{b}\right]\right]$ (6)
 > $R≔\mathrm{PolynomialRing}\left(\left[a,b,c\right]\right):$
 > $\mathrm{dec1}≔\mathrm{map}\left(\mathrm{op},\mathrm{map}\left(\mathrm{RealTriangularize},S,R\right)\right)$
 ${\mathrm{dec1}}{≔}\left[{\mathrm{regular_semi_algebraic_system}}{,}{\mathrm{regular_semi_algebraic_system}}{,}{\mathrm{regular_semi_algebraic_system}}{,}{\mathrm{regular_semi_algebraic_system}}\right]$ (7)
 > $\mathrm{dec2}≔\mathrm{RemoveRedundantComponents}\left(\mathrm{dec1},R\right)$
 ${\mathrm{dec2}}{≔}\left[{\mathrm{regular_semi_algebraic_system}}\right]$ (8)
 > $\mathrm{evalb}\left(\mathrm{nops}\left(\mathrm{dec2}\right)<\mathrm{nops}\left(\mathrm{dec1}\right)\right)$
 ${\mathrm{true}}$ (9)
 > $\mathrm{IsContained}\left(\mathrm{dec1},\mathrm{dec2},R\right)$
 ${\mathrm{true}}$ (10)
 > $\mathrm{IsContained}\left(\mathrm{dec1},\mathrm{dec2},R\right)$
 ${\mathrm{true}}$ (11)

References

 Xie, Y. "Fast Algorithms, Modular Methods, Parallel Approaches and Software Engineering for Solving Polynomial Systems Symbolically" Ph.D. Thesis, University of Western Ontario, Canada, 2007.

Compatibility

 • The RegularChains[SemiAlgebraicSetTools][RemoveRedundantComponents] command was introduced in Maple 16.
 • The lrsas parameter was introduced in Maple 16.