On the bit complexity of finding points in connected components of a smooth real hypersurface

Abstract

We present a full analysis of the bit complexity of an efficient algorithm for the computation of at least one point in each connected component of a smooth real hypersurface. This is a basic and important operation in semi-algebraic geometry: it gives an upper bound on the number of connected components of a real hypersurface, and is also used in many higher level algorithms. Our starting point is an algorithm by Safey El Din and Schost (Polar varieties and computation of one point in each connected component of a smooth real algebraic set, ISSAC'03). This algorithm uses random changes of variables that are proved to generically ensure certain desirable geometric properties. The cost of the algorithm was given in an algebraic complexity model; the analysis of the bit complexity and the error probability were left for future work. Our paper answers these questions. Our main contribution is a quantitative analysis of several genericity statements, such as Thom's weak transversality theorem or Noether normalization properties for polar varieties.