  Let V be a smooth equidimensional quasi-affine variety of dimension r over
the complex numbers $C$ and let $F$ be a $(p\times s)$-matrix of coordinate
functions of $C[V]$, where $s\ge p+r$. The pair $(V,F)$ determines a vector
bundle $E$ of rank $s-p$ over $W:=\{x\in V:\mathrm{rk} F(x)=p\}$. 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 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.
