RegularChains[ChainTools] - Maple Programming Help

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RegularChains[ChainTools]

 Regularize
 make a polynomial regular or null with respect to a regular chain

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

Parameters

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

Description

 • The command Regularize(p, rc, R) returns a list made of two lists. The first one consists of regular chains $\mathrm{reg_i}$ such that p is regular modulo the saturated ideal of $\mathrm{reg_i}$. The second one consists of regular chains $\mathrm{sing_i}$ such that p is null modulo the saturated ideal of $\mathrm{sing_i}$.
 • In addition, the union of the regular chains of these lists is a decomposition of rc in the sense of Kalkbrener.
 • If 'normalized'='yes' is passed, all the returned regular chains are normalized.
 • If 'normalized'='strongly' is passed, all the returned regular chains are strongly normalized.
 • If 'normalized'='yes' is present, rc must be normalized.
 • If 'normalized'='strongly' is present, rc must be strongly normalized.
 • The command RegularizeDim0 implements another algorithm with the same purpose as that of the command Regularize. However it is specialized to zero-dimensional regular chains in prime characteristic. When both algorithms apply, the latter usually outperforms the former one.
 • This command is part of the RegularChains[ChainTools] package, so it can be used in the form Regularize(..) 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][Regularize](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$$\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,z\right]\right)$
 ${R}{:=}{\mathrm{polynomial_ring}}$ (1)
 > $\mathrm{rc}≔\mathrm{Empty}\left(R\right)$
 ${\mathrm{rc}}{:=}{\mathrm{regular_chain}}$ (2)
 > $\mathrm{rc}≔\mathrm{Chain}\left(\left[z\left(z-1\right),y\left(y-2\right)\right],\mathrm{rc},R\right);$$\mathrm{Equations}\left(\mathrm{rc},R\right)$
 ${\mathrm{rc}}{:=}{\mathrm{regular_chain}}$
 $\left[{{y}}^{{2}}{-}{2}{}{y}{,}{{z}}^{{2}}{-}{z}\right]$ (3)
 > $p≔zx+y$
 ${p}{:=}{x}{}{z}{+}{y}$ (4)
 > $\mathrm{reg},\mathrm{sing}≔\mathrm{op}\left(\mathrm{Regularize}\left(p,\mathrm{rc},R\right)\right)$
 ${\mathrm{reg}}{,}{\mathrm{sing}}{:=}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]{,}\left[{\mathrm{regular_chain}}\right]$ (5)
 > $\mathrm{map}\left(\mathrm{Equations},\mathrm{reg},R\right)$
 $\left[\left[{y}{-}{2}{,}{z}\right]{,}\left[{y}{,}{z}{-}{1}\right]{,}\left[{y}{-}{2}{,}{z}{-}{1}\right]\right]$ (6)
 > $\mathrm{map}\left(\mathrm{Equations},\mathrm{sing},R\right)$
 $\left[\left[{y}{,}{z}\right]\right]$ (7)
 > $\left[\mathrm{seq}\left(\mathrm{SparsePseudoRemainder}\left(p,{\mathrm{reg}}_{i},R\right),i=1..\mathrm{nops}\left(\mathrm{reg}\right)\right)\right]$
 $\left[{2}{,}{x}{,}{2}{+}{x}\right]$ (8)
 > $\mathrm{seq}\left(\mathrm{SparsePseudoRemainder}\left(p,{\mathrm{sing}}_{i},R\right),i=1..\mathrm{nops}\left(\mathrm{sing}\right)\right)$
 ${0}$ (9)