The command DefiningSet(rc, d, R) returns the defining set of rc with respect to the last d variables, regarded as parameters. This is a constructible set .

Given a positive integer d, the regular chain rc can be split into two parts. Denote by  the set of the polynomials in rc involving only the last d variables, and denote by  the other polynomials of rc. Certainly, both  and  are regular chains.

Let  be the quasi-component of . For a point  in , after specializing  at , two situations arise:

(1) either  is not a regular chain anymore;

(2) or  is still a regular chain.

There is a subtle point: after specializing  at , it might happen that it is still a regular chain, but its shape changes. In other words, the degree of the geometric object given by  could change. The term specialize well, defined below, takes these cases into account.

The regular chain rc specializes well at a point  of  if  is a regular chain after specialization and no initial of polynomials in rc1 vanish during the specialization.

The defining set of rc with respect to the last d variables consists of the points in  at which rc specializes well.

This command is part of the RegularChains[ParametricSystemTools] package, so it can be used in the form DefiningSet(..) only after executing the command with(RegularChains[ParametricSystemTools]). However, it can always be accessed through the long form of the command by using RegularChains[ParametricSystemTools][DefiningSet](..).