Solving determinantal systems using homotopy techniques

Abstract

Let K be a field of characteristic zero and let K‾ be an algebraic closure of K. Consider a sequence of polynomials G=(g1,…,gs) in K[X1,…,Xn] with s<n, a polynomial matrix F=[fi,j]∈K[X1,…,Xn]p×q, with p≤q and n=q−p+s+1, and the algebraic set Vp(F,G) of points in K‾ at which all polynomials in G and all p-minors of F vanish. Such polynomial systems appear naturally in polynomial optimization or computational geometry.We provide bounds on the number of isolated points in Vp(F,G) depending on the maxima of the degrees in rows (resp. columns) of F and we design probabilistic homotopy algorithms for computing those points. These algorithms take advantage of the determinantal structure of the system defining Vp(F,G). In particular, the algorithms run in time that is polynomial in the bound on the number of isolated points.