Solving determinantal systems using homotopy techniques

Let $\text{K}$ be a field of characteristic zero and $\text{Kbar}$ be an algebraic closure of $\text{K}$. Consider a sequence of polynomials$G=(g\mathrm{\_}1,\dots ,g\mathrm{\_}s)$ in $\text{K}[X\mathrm{\_}1,\dots ,X\mathrm{\_}n]$, a polynomial matrix $\F=[f_{i,j}] \in \K[X_1,\dots,X_n]^{p \times q}$,with$p \leq q$,andthealgebraicset$V_p(F, G)$ofpointsin$\KKbar$atwhichallpolynomialsin$\G$andall$p$-minorsof$\F$vanish. Such polynomial systems appear naturally in e.g. polynomial optimization, computational geometry.We provide bounds on the number of isolated points in $V\mathrm{\_}p(F,G)$ depending on the maxima of the degrees in rows (resp. columns) of $\text{F}$. Next, we design homotopy algorithms for computing those points. These algorithms take advantage of the determinantal structure of the system defining $V\mathrm{\_}p(F,G)$. In particular, the algorithms run in time that is polynomial in the bound on the number of isolated points.