Betti Number Bounds, Applications and Algorithms

Summary

Topological complexity of semialgebraic sets in ℝ^k has been studied by many researchers over the past fifty years. An important measure of the topological complexity are the Betti numbers. Quantitative bounds on the Betti numbers of a semialgebraic set in terms of various parameters (such as the number and the degrees of the polynomials defining it, the dimension of the set etc.) have proved useful in several applications in theoretical computer science and discrete geometry. The main goal of this survey paper is to provide an up to date account of the known bounds on the Betti numbers of semialgebraic sets in terms of various parameters, sketch briefly some of the applications, and also survey what is known about the complexity of algorithms for computing them.

1. Introduction

Let R be a real closed field and S a semialgebraic subset of R^k, defined by a Boolean formula, whose atoms are of the form P = 0, P > 0 , P < 0 , where P ∈ p for some finite family of polynomials P ⊂ R[X₁,…, X _k . It is well known [Bochnak et al. 1987] that such sets are finitely triangulable. Moreover, if the cardinality of P and the degrees of the polynomials in P are bounded, then the number of topological types possible for S is finite [Bochnak et al. 1987]. (Here, two sets have the same topological type if they are semialgebraically homeomorphic). A natural problem then is to bound the topological complexity of S in terms of the various parameters of the formula defining S.