Abstract. In this paper we prove tight bounds on the combinatorial and topological complexity of sets defined in terms of n definable sets belonging to some fixed definable family of sets in an o-minimal structure. This generalizes the combinatorial parts of similar bounds known in the case of semi-algebraic and semi-Pfaffian sets, and as a result vastly increases the applicability of results on combinatorial and topological complexity of arrangements studied in discrete and computational geometry. As a sample application, we extend a Ramsey-type theorem due to Alon et al. [3], originally proved for semialgebraic sets of fixed description complexity to this more general setting. 1.

In this paper we consider the problem of bounding the Betti numbers, bi(S), of a semi-algebraic set S ⊂ R k defined by polynomial inequalities P1 ≥ 0,..., Ps ≥ 0, where Pi ∈ R[X1,..., Xk] , s < k, and deg(Pi) ≤ 2, for 1 ≤ i ≤ s. We prove that for 0 ≤ i ≤ k − 1, bi(S) ≤ 1 1 + (k − s) +

a Boolean formula without negations, with atoms P = 0, P ≥ 0, P ≤ 0, P ∈ P ∪ Q. We prove that the sum of the Betti numbers of S is bounded by (ℓsmd) O(m+k). This is a common generalization of previous results in [7] and [2] on bounding the Betti numbers of closed semi-algebraic sets defined by polynomials of degree d and 2, respectively. We also describe algorithms for computing the Euler-Poincaré characteristic, as well as all the Betti numbers of such sets, generalizing similar algorithms described in [7, 4] and [5]. The complexity of the first algorithm is bounded by (ℓsmd) O(m(m+k)) , while that of the second is bounded by (ℓsmd) 2O(m+k).

Abstract. 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.

Abstract. Computing various topological invariants of semi-algebraic sets in single exponential time is an active area of research. Several algorithms are known for deciding emptiness, computing the number of connected components of semi-algebraic sets in single exponential time etc. However, an algorithm for computing all the Betti numbers of a given semi-algebraic set in single exponential time is still lacking. In this paper we describe a new, improved algorithm for computing the Euler-Poincaré characteristic (which is the alternating sum of the Betti numbers) of the realization of each realizable sign condition of a family of polynomials restricted to a real variety. The complexity of the algorithm is sk ′ +1O(d) k +sk ′ ((k ′ log2(s)+k log2(d))d) O(k) where s is the number of polynomials, k the number of variables, d a bound on the degrees, and k ′ the real dimension of the variety. A consequence of our result is that the Euler-Poincaré characteristic of any locally closed semi-algebraic set can be computed with the same complexity. The best complexity of any previously known single exponential time algorithm for computing the Euler-Poincaré characteristic of semi-algebraic sets worked only for a more restricted class of closed semi-algebraic sets and had a complexity of (ksd) O(k). Keywords. Semi-algebraic sets, Euler-Poincaré characteristic

Abstract. In this lecture we introduce semi-algebraic sets, Tarski-Seidenberg principle, give basic definitions of homology and co-homology groups of semi-algebraic sets, and state certain quantitative results which give tight bounds on the ranks of these groups. We also state several

In this paper we prove an asymptotically tight bound (asymptotic with respect to the number of polynomials for fixed degrees and number of variables) on the number of connected components of the realizations of all realizable sign conditions of a family of real polynomials. More precisely, we prove that the number of connected components of the realizations of all realizable sign conditions of a family of s polynomials in R[X1,..., Xk] whose degrees are at most d, is bounded by (2d) k k! sk + O(s k−1). This improves the best upper bound known previously, which was

We prove Pach-Sharir type incidence theorems for a class of curves in R n and surfaces in R 3, which we call pseudoflats. In particular, our results apply to a wide class of generic irreducible real algebraic sets of bounded degree. 1

the Euler–Poincaré characteristic of semi-algebraic sets defined by partly quadratic systems of polynomials

complexity in o-minimal geometry