in principle, $\widehat f$ and $\real^n/G$.

The orbit space of a compact group can be realized as a connected semi-algebraic subset ([*i.e.*]{}, a subset defined by algebraic equalities and inequalities) of a Euclidean space $\real^q$. It turns out to be formed by the union of connected semialgebraic manifolds of different dimensions ([*primary strata*]{}). The images of $G$-orbits with the same orbit–type form [*isotropy–type strata*]{}, whose connected components are primary strata.

The properties of a function defined on $\real^n/G$, may critically depend on the geometry of this space, that has to be known explicitly. A simple way to obtain a substantial determination of the algebraic equations and inequalities defining the range of an orbit map and its strata has been suggested in [@020; @020a; @020b] (see also [@651; @650]). These relations can be expressed in the form of positivity and rank conditions of a matrix $\widehat P(p)$, whose elements are polynomial functions of $p\in\real^q$. In this way, the defining relations of the strata are obtained in the form of algebraic equations and inequalities.

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 