Degeneracy Loci and Polynomial Equation Solving

Let V be a smooth, equidimensional, quasi-affine variety of dimension r over C , and let F be a ( p × s ) matrix of coordinate functions of C [ V ] , where s ≥ p + r . The pair ( V , F ) determines a vector bundle E of rank s - p over W : = { x ∈ V ∣ rk F ( x ) = p } . We associate with ( V , F ) a descending chain of degeneracy loci of E (the generic polar varieties of V 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.