The command Regularize(p, rc, R) returns a list made of two lists. The first one consists of regular chains  such that p is regular modulo the saturated ideal of . The second one consists of regular chains  such that p is null modulo the saturated ideal of .

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](..).