Polar, bipolar and copolar varieties: Real solving of algebraic varieties with intrinsic complexity

Abstract

This survey covers a decade and a half of joint work with L. Lehmann, G. M. Mbakop, and L. M. Pardo. We address the problem of finding a smooth algebraic sample point for each connected component of a real algebraic variety, being only interested in components which are generically smooth locally complete intersections. The complexity of our algorithms is essentially polynomial in the degree of suitably defined generalized polar varieties and is therefore intrinsic to the problem under consideration.