Degeneracy loci and polynomial equation solving 1 Bernd Bank

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. Research partially supported by the following Argentinian, French and Spanish grants: CONICET Res 4541-12, PIP 11220090100421 CONICET, UBACYT 20020100100945 and 20020110100063, PICT–2010–0525, Digiteo DIM 2009–36HD “MaGiX” grant of the Région Ile-deFrance, ANR-2010-BLAN-0109-04 “LEDA”, MTM2010-16051. Humboldt-Universität zu Berlin, Institut für Mathematik, 10099 Berlin, Germany. [email protected] Laboratoire d’informatique LIX, UMR 7161 CNRS, campus de l’École polytechnique, 91128 Palaiseau Cedex, France. [email protected] Departamento de Computación, Universidad de Buenos Aires and CONICET, Ciudad Univ., Pab.I, 1428 Buenos Aires, Argentina, and Departamento de Matemáticas, Estad́ıstica y Computación, Facultad de Ciencias, Universidad de Cantabria, 39071 Santander, Spain. [email protected] Laboratoire d’informatique LIX, UMR 7161 CNRS, campus de l’École polytechnique, 91128 Palaiseau Cedex, France. [email protected] Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento and CONICET, J. M. Gutierrez 1150 (B1613GSX) Los Polvorines, Prov. de Buenos Aires, Argentina. [email protected] Instituto Matemático Luis Santaló CONICET and Departamento de Matemáticas, Facultad de Ciencias Exactas y Naturales, UBA, 1428 Buenos Aires, Argentina. [email protected]