following, we shall identify orbit spaces and their strata with their images through orbit maps.

If $G$ is coregular, the $p_i$’s, with range in the semialgebraic set $\Delta$ defined by the inequalities $\widehat P(p)>0$ (and no equation!) provide a one-to-one parametrization of the principal stratum of $\real^n/G$. If the stratum is singular, the $p_i$’s are not independent parameters, being bounded also by the equations defining the stratum. In applications or, simply, to get a better understanding of the geometry of the stratum (its connectivity properties, its boundary, etc. ...), one often needs to solve explicitly these equations, which are obtained in implicit form from the theorem recalled above. Often, this is very difficult to do using standard algorithms. In the following section, we propose a general method to derive the equations defining the singular strata of the orbit space of any CLG in the form of explicit rational parametric relations.

The mathematical apparatus just recalled offers effective tools, for instance, in the determination of possible patterns of spontaneous symmetry breaking, when the ground state of the system is determined by the absolute minimum of an invariant potential. Let us recall the basic ideas. In this context, the vector $x\in
\real^n$ 