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.

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

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

$R≔\mathrm{polynomial\_ring}$

$\mathrm{rc1}≔\mathrm{Chain}\left(\left[y\left(y+1\right)\right]\,\mathrm{Empty}\left(R\right)\,R\right)$

$\mathrm{rc1}≔\mathrm{regular\_chain}$

$\mathrm{rc2}≔\mathrm{Chain}\left(\left[x\,y\right]\,\mathrm{Empty}\left(R\right)\,R\right)$

$\mathrm{rc2}≔\mathrm{regular\_chain}$

$\mathrm{out}≔\mathrm{RemoveRedundantComponents}\left(\left[\mathrm{rc1}\,\mathrm{rc2}\right]\,R\right)$

$\mathrm{out}≔\left[\mathrm{rc1}\right]$

$\mathrm{map}\left(\mathrm{Equations}\,\mathrm{out}\,R\right)$

$\left[\left[y^2+y\right]\right]$

$\mathrm{C1}≔\left[0<a\&comma;0<b\&comma;0<c\&comma;a<b+c\&comma;b<a+c\&comma;c<a+b\&comma;b^2+a^2-c^2\le 0\right]\&colon;$

$\mathrm{C2}≔\left[0<a\&comma;0<b\&comma;0<c\&comma;a<b+c\&comma;b<a+c\&comma;c<a+b\&comma;c{\left(b^2+a^2-c^2\right)}^2<a{b}^2\left(2ac-\left(c^2+a^2-b^2\right)\right)\right]\&colon;$

$\mathrm{C3}≔\left[a-c<0\&comma;0<a\&comma;0<b\&comma;0<c\&comma;a<b+c\&comma;b<a+c\&comma;c<a+b\right]\&colon;$

$S≔\left[\mathrm{C1}\&comma;\mathrm{C2}\&comma;\mathrm{C3}\right]$

$S≔\left[\left[0<a\&comma;0<b\&comma;0<c\&comma;a<b+c\&comma;b<a+c\&comma;c<a+b\&comma;a^2+b^2-c^2\le 0\right]\&comma;\left[0<a\&comma;0<b\&comma;0<c\&comma;a<b+c\&comma;b<a+c\&comma;c<a+b\&comma;c{\left(a^2+b^2-c^2\right)}^2<a{b}^2\left(-a^2+2ac+b^2-c^2\right)\right]\&comma;\left[a-c<0\&comma;0<a\&comma;0<b\&comma;0<c\&comma;a<b+c\&comma;b<a+c\&comma;c<a+b\right]\right]$

$R≔\mathrm{PolynomialRing}\left(\left[a\&comma;b\&comma;c\right]\right)\&colon;$

$\mathrm{dec1}≔\mathrm{map}\left(\mathrm{op}\&comma;\mathrm{map}\left(\mathrm{RealTriangularize}\&comma;S\&comma;R\right)\right)$

$\mathrm{dec1}≔\left[\mathrm{regular\_semi\_algebraic\_system}\&comma;\mathrm{regular\_semi\_algebraic\_system}\&comma;\mathrm{regular\_semi\_algebraic\_system}\&comma;\mathrm{regular\_semi\_algebraic\_system}\right]$

$\mathrm{dec2}≔\mathrm{RemoveRedundantComponents}\left(\mathrm{dec1}\&comma;R\right)$

$\mathrm{dec2}≔\left[\mathrm{regular\_semi\_algebraic\_system}\right]$

$\mathrm{evalb}\left(\mathrm{nops}\left(\mathrm{dec2}\right)<\mathrm{nops}\left(\mathrm{dec1}\right)\right)$

$\mathrm{true}$

$\mathrm{IsContained}\left(\mathrm{dec1}\&comma;\mathrm{dec2}\&comma;R\right)$

$\mathrm{true}$

$\mathrm{IsContained}\left(\mathrm{dec1}\&comma;\mathrm{dec2}\&comma;R\right)$

$\mathrm{true}$

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.

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

The lrsas parameter was introduced in Maple 16.

For more information on Maple 16 changes, see Updates in Maple 16.