  Let $\K$ be a field of characteristic zero and $\Kbar$ be an algebraic
closure of $\K$. Consider a sequence of polynomials$G=(g\_1,\dots,g\_s)$ in
$\K[X\_1,\dots,X\_n]$, a polynomial matrix $\F=[f\_{i,j}] \in
\K[X\_1,\dots,X\_n]^{p \times q}$, with $p \leq q$,and the algebraic set
$V\_p(F, G)$ of points in $\KKbar$ at which all polynomials in $\G$ and all
$p$-minors of $\F$vanish. Such polynomial systems appear naturally in e.g.
polynomial optimization, computational geometry.We provide bounds on the number
of isolated points in $V\_p(F, G)$ depending on the maxima of the degrees in
rows (resp. columns) of $\F$. Next, we design homotopy algorithms for computing
those points. These algorithms take advantage of the determinantal structure of
the system defining $V\_p(F, G)$. In particular, the algorithms run in time
that is polynomial in the bound on the number of isolated points.
