On the geometry of polar varieties

Bernd Bank , Marc Giusti , Joos Heintz , Mohab Safey El Din , Eric Schost

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 2009-10

MSC 2000

14P05 Real algebraic sets

14B05 Singularities

14Q10 Surfaces, hypersurfaces

14Q15 Higher-dimensional varieties

68W30 Symbolic computation and algebraic computation

Abstract
The aim of this paper is a comprehensive presentation of the geometrical tools which are necessary to prove the correctness of several up to date algorithms with intrinsic complexity bounds for the problem of real root nding in (mainly) smooth real algebraic varieties given by reduced regular sequences of polynomial equations (see [4, 5, 6, 7], [30] and [31]). The results exposed in this paper form also the geometrical main ingredients for the computational treatment of singular hypersurfaces (see [8]). In particular, we show the non{emptiness of suitable fully generic dual polar varieties of (possibly singular) real varieties, show that fully generic polar varieties may become singular at smooth points of the original variety, give a sucient criterion when this is not the case, give an intrinsic degree estimate for polar varieties and introduce the new concept of a suciently generic polar variety. Our statements are illustrated by examples and a computer experiment.