$v_\bot\in V_\bot$: $v=(x_1,\dots ,x_\nu)$, $v_\bot=(x_{\nu+1},\dots ,x_n)$. Then, a $G$-invariant polynomial $f(x)$, can be thought of as a polynomial in $v$ and $v_\bot$ and it is easy to realize that it cannot contain linear terms in $v_\bot$, being $v$ invariant under $H$. As a consequence,

[$${\partial_i f(x)=0,\ {\rm for\ }x\in V\ {\rm and\
}i=\nu+1,\dots ,n.}\label{3}$$]{} Now, starting from the very definition of $P(x)$ (see [(\[matP\])]{}), for every $x=v\in V$ we obtain, using [(\[3\])]{} and the identity $p(v)=\phi(\lambda(v))$, [$${\begin{array}{rcl}
\widehat P_{ab}(p(v))&=&\sum_{i=1}^\nu\,\partial_i p_a(v)\,\partial_i p_b(v)\\
&&\\
&=& \left.
\sum_{\alpha,\beta=1}^l\,\partial_\alpha\phi_a(\lambda)\,
\partial_\beta\phi_b(\lambda)
\right|_{\lambda=\lambda(v)}\,\sum_{i=1}^\nu\,\partial_i
\lambda_\alpha(v)\,
\partial_i\lambda_\beta(v)\\
&&\\
&=& \left.
\left(J(\lambda)\,\widehat\Lambda(\lambda)\,J(\lambda)^{\rm
T}\right|_{\lambda=\lambda(v)}\right)_{ab} ,
\end{array}}\label{5}$$]{} where, the superscript $T$ denotes transposition and $J(\lambda)$ is the Jacobian matrix of the transformation $p=\phi(\lambda)$:

[$${J_{a\alpha}(\lambda)=\partial_a\phi_\alpha (\lambda),\qquad
a=1,\dots ,q,\ \ \alpha=1,\dots ,l.}\label{7}$$]{} So, we can conclude that, for all $\lambda\in V/K$,

[$${\widehat P(\phi(\lambda))=
J(\lambda)\,\widehat\Lambda(\lambda)\,J(\lambda)^{\rm T}.}\label{6}$$]{}

Equation [(\[6\])]{} leads to an easy calculation of the points $\lambda\in \widehat{\mathcal V}$, that is, of the points $\lambda\in \widehat\Sigma$ whose image $p=\phi(\lambda)$ is in $\widehat S$. In fact, from [(\[rango\])]{}, these points are characterized by the conditions $\lambda\in\widehat\Sigma$ and rank($\widehat P(\phi(\lambda)))=l$. Since, for $\lambda\in\widehat\Sigma$, the matrix $\widehat
\Lambda(\lambda)$ is positive definite with rank $l$, we can conclude that the range of $\lambda$ assuring the location of the 