RegularChains[ChainTools] - Maple Help

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RegularChains[ChainTools]

 ListConstruct
 constructs regular chains

 Calling Sequence ListConstruct(lp, rc, R) ListConstruct(p, rc, R, 'normalized'='yes') ListConstruct(p, rc, R, 'normalized'='strongly')

Parameters

 lp - list of polynomials of R rc - regular chain of R R - polynomial ring 'normalized'='yes' - (optional) boolean flag 'normalized'='strongly' - (optional) boolean flag

Description

 • The command ListConstruct(lp, rc, R) returns a list of regular chains ${\mathrm{rc}}_{i}$ which form a triangular decomposition of the regular chain obtained by extending rc with lp.
 • It is assumed that lp is a list of non-constant polynomials sorted in increasing main variable, and that any main variable of a polynomial in lp is strictly greater than any algebraic variable of rc.
 • It is also assumed that the polynomials of rc together with those of lp form a regular chain.
 • Although rc with lp is assumed to form a regular chain, several regular chains may be returned; this is because the polynomials of lp may be factorized with respect to rc.
 • To avoid these possible factorizations, use RegularChains[ChainTools][Chain]
 • If 'normalized'='yes' is present, then rc must be normalized. In addition, every returned regular chain is normalized.
 • If 'normalized'='strongly' is present, then rc must be strongly normalized. In addition, every returned regular chain is strongly normalized.
 • This command is part of the RegularChains[ChainTools] package, so it can be used in the form ListConstruct(..) 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][ListConstruct](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[t,x,y,z\right]\right)$
 ${R}{:=}{\mathrm{polynomial_ring}}$ (1)
 > $\mathrm{py}≔{y}^{2}+2y+1$
 ${\mathrm{py}}{:=}{{y}}^{{2}}{+}{2}{}{y}{+}{1}$ (2)
 > $\mathrm{pt}≔{t}^{2}+y$
 ${\mathrm{pt}}{:=}{{t}}^{{2}}{+}{y}$ (3)
 > $\mathrm{rc}≔\mathrm{Empty}\left(R\right)$
 ${\mathrm{rc}}{:=}{\mathrm{regular_chain}}$ (4)
 > $\mathrm{lrc}≔\mathrm{ListConstruct}\left(\left[\mathrm{py},\mathrm{pt}\right],\mathrm{rc},R\right)$
 ${\mathrm{lrc}}{:=}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (5)
 > $\mathrm{map}\left(\mathrm{Equations},\mathrm{lrc},R\right)$
 $\left[\left[{t}{-}{1}{,}{y}{+}{1}\right]{,}\left[{t}{+}{1}{,}{y}{+}{1}\right]\right]$ (6)