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](..).

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

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

$R≔\mathrm{polynomial\_ring}$

$\mathrm{rc}≔\mathrm{Empty}\left(R\right)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\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)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\left[y^2-2y\,z^2-z\right]$

$p≔zx+y$

$p≔zx+y$

$\mathrm{reg},\mathrm{sing}≔\mathrm{op}\left(\mathrm{Regularize}\left(p\,\mathrm{rc}\,R\right)\right)$

$\mathrm{reg},\mathrm{sing}≔\left[\mathrm{regular\_chain}\,\mathrm{regular\_chain}\,\mathrm{regular\_chain}\right],\left[\mathrm{regular\_chain}\right]$

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

$\left[\left[y-2\,z\right]\,\left[y\,z-1\right]\,\left[y-2\,z-1\right]\right]$

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

$\left[\left[y\,z\right]\right]$

$\left[\mathrm{seq}\left(\mathrm{SparsePseudoRemainder}\left(p\,\mathrm{reg}\left[i\right]\,R\right)\,i=1..\mathrm{nops}\left(\mathrm{reg}\right)\right)\right]$

$\left[2\,x\,x+2\right]$

$\mathrm{seq}\left(\mathrm{SparsePseudoRemainder}\left(p\,\mathrm{sing}\left[i\right]\,R\right)\,i=1..\mathrm{nops}\left(\mathrm{sing}\right)\right)$

$0$