$\real^n/G$ , $i=1,2$, then $H_1<H_2$ iff $\widehat S_2$ is in the boundary of $\widehat S_1$; therefore, more peripheral strata of $\real^n/G$ have greater orbit–type. The number of distinct orbit–types of $G$ is finite and there is a unique minimum orbit–type, the [*principal orbit–type*]{}, corresponding to the principal stratum.

The ring $\real [x]^G$ of real $G$-invariant polynomial functions of $x$ is finitely generated [@286; @286a; @286b; @Nag]. There exists, therefore, a finite minimal collection of homogeneous $G$-invariant polynomials $p(x) = ( p_1(x), p_{2}(x), \ldots,
p_{q}(x) )$ (minimal integrity basis for the ring of $G$-invariant polynomials, henceforth abbreviated in MIB) such that any element $F\in\real[x]^G$ can be expressed as a polynomial function $\widehat F$ of $p(x)$:

[$${\widehat{F}(p(x)) \,=\, F(x), \, \forall x \in
\real^{n}\,.}\label{e2}$$]{} The number $q$ of elements of a minimal integrity basis and their homogeneity [*degrees*]{} $d_i$’s are only determined by the group $G$.

The group $G$ is said to be [*coregular*]{} if the elements of its MIB’s are algebraically independent. The elements of a MIB of a non coregular group satisfy a set of algebraic identities in $\real^n$: $\widehat F_A(p(x))\,=\,0$, $A=1,\dots$ and the associated set of equations

[$${ \widehat F_A(p)\,=\,0,\qquad A=1,\dots\,,}\label{e5}$$]{} defines an algebraic variety in $\real^q$, which is 