The command Construct(p, rc, R) returns a list of regular chains  which form a triangular decomposition of the regular chain obtained by extending rc with p.

This assumes that p is a non-constant with main variable greater than any algebraic variable of rc, and that the initial of p is regular modulo the saturated ideal of rc. Hence p and rc form together a regular chain.

Although rc with p is assumed to form a regular chain, several regular chains may be returned; this is because the polynomial p may be factorized with respect to rc in order to simplify the expressions in the regular chains .

Such factorizations will happen if they can be performed quickly. For instance, if p involves only one variable.

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 Construct(..) 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][Construct](..).