Degeneracy loci and polynomial equation solving

Bernd Bank , Marc Giusti , Joos Heintz , Gregoire Lecerf , Guillermo Matera , Pablo Solerno

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 2013-08

MSC 2000

14M10 Complete intersections

14M12 Determinantal varieties

14Q20 Effectivity

14P05 Real algebraic sets

68W30 Symbolic computation and algebraic computation

Abstract
Let $V$ be a smooth equidimensional quasi-affine variety of dimension $r$ over $\C$ 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 $W:=\{x\in V | \rk F(x)=p\}$. 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.