point $p=\phi(\lambda)$ in $\widehat S$ is the semialgebraic subset $\Delta$ of $\real^l$, determined by the following inequalities:

[$${\Delta=\{\lambda\in\real^l\mid \widehat{\Lambda}(\lambda)>0
\ {\rm and}\ {\rm rank}(J(\lambda))=l\}.}\label{8}$$]{} For $\lambda\in\Delta$, the relation $p=\phi(\lambda)$ yields a global rational parametrization for $\widehat S$.

It will be worthwhile to stress that the parametrization we have suggested is, in some way, canonical: the unique arbitrariness in the choice of the parameters is related to the choice of the MIB’s.

An important byproduct of the result just proved is a simple test of the connection of $\widehat S$, that could be difficult or impossible to read directly from the equations of the stratum in implicit form. In fact, the condition that the boundary of $\overline{\widehat S}$ coincides with the boundary of $\widehat
S$ is equivalent to the condition rank($J(\lambda))=l$ for all $\lambda\in\widehat \Sigma$.

If $K$ is not coregular, there are polynomial relations $F_A(\lambda)=0$ among the elements of the MIB $(\lambda_1,\dots
,\lambda_l)$ and $l>$dim$(\widehat S)$. As already stressed, to obtain a one-to-one parametrization of the points of $\widehat S$ by means of the $\lambda_\alpha$’s, one has to eliminate the redundant parameters by solving the equations $F_A(\lambda)=0$. This may be feasible, but only locally and the resulting parametrization will not be 