A G ] 1 6 D ec 2 01 3 Degeneracy loci and polynomial equation solving 1

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