Solving determinantal systems using homotopy techniques

Abstract

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 $p\le q$,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.