Degeneracy Loci and Polynomial Equation Solving

Abstract

Let (Formula presented.) be a smooth, equidimensional, quasi-affine variety of dimension (Formula presented.) over (Formula presented.), and let (Formula presented.) be a (Formula presented.) matrix of coordinate functions of (Formula presented.), where (Formula presented.). The pair (Formula presented.) determines a vector bundle (Formula presented.) of rank (Formula presented.) over (Formula presented.). We associate with (Formula presented.) a descending chain of degeneracy loci of (Formula presented.) (the generic polar varieties of (Formula presented.) represent a typical example of this situation). The maximal degree of these degeneracy loci constitutes the essential ingredient for the uniform, bounded-error probabilistic pseudo-polynomial-time algorithm that we will design and that solves a series of computational elimination problems that can be formulated in this framework. We describe applications to polynomial equation solving over the reals and to the computation of a generic fiber of a dominant endomorphism of an affine space.