gradients of the elements of the MIB $\{p(x)\}$:

$$\label{matP}
P_{ab}(x) \,=\, \sum_{i=1}^n \,\partial_i p_a(x)\,
\partial_i p_b(x) \,=\, \widehat{P}_{ab}(p(x)), \hspace{3em}
a,b = 1, \ldots, q,$$

where in the last member, use has been made of Hilbert’s theorem, in order to express $P_{ab}(x)$ as a polynomial function of $p_1(x), \dots ,p_q(x)$.

The following theorem [@020; @020a; @020b] (see also [@651]) clarifies the meaning and points out the role of the matrix $\widehat{P}(p)$:

\[T1\] Let $G$ be a compact subgroup of the real group O(n), $p$ the orbit map $\real^n\rightarrow\real^q$ defined by the MIB $(p_1(x),p_2(x),\ldots ,p_q(x))$ and $\widehat{P}(p)$ the matrix defined in [(\[matP\])]{}. Then the range $p(\real^{n})$ of $p$ is the unique semialgebraic connected subset of the variety ${\cal
Z}\subseteq \real^q$ of the relations among the elements of the MIB where $\widehat{P}(p)$ is positive semi-definite. The $k$–dimensional primary strata of $p(\real^{n})$ are the connected components of the set $\{ p \in {\cal Z} \mid
\widehat{P}(p) \geq 0, \, {\rm rank} (\widehat{P}(p)) = k \}$; they are the images of the connected components of the k–dimensional isotropy–type strata of $\real^{n}/G$. In particular, the set of interior points of $p(\real^{n})$, where $\widehat{P}(p)$ has the maximum rank, is the image of the principal stratum.

In the 