D ec 2 01 3 Degeneracy loci and polynomial equation solving 1

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 | rkF (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 which we are going to design and which 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.