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14
Combinatorial complexity in ominimal geometry
, 2009
"... 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 ominimal structure. This generalizes the combinatorial parts of similar bounds known in the case of semialgebraic ..."
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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 ominimal structure. This generalizes the combinatorial parts of similar bounds known in the case of semialgebraic and semiPfaffian 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 Ramseytype theorem due to Alon et al. [3], originally proved for semialgebraic sets of fixed description complexity to this more general setting.
A sharper estimate on the Betti numbers of sets defined by quadratic inequalities
 Discrete and Computational Geometry, to appear. SAUGATA BASU, DMITRII V. PASECHNIK, AND MARIEFRANÇOISE
"... In this paper we consider the problem of bounding the Betti numbers, bi(S), of a semialgebraic 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) + ..."
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Cited by 9 (6 self)
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In this paper we consider the problem of bounding the Betti numbers, bi(S), of a semialgebraic 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) +
Betti Number Bounds, Applications and Algorithms
 Current Trends in Combinatorial and Computational Geometry: Papers from the Special Program at MSRI, MSRI Publications Volume 52
, 2005
"... 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 ..."
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Cited by 9 (6 self)
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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.
Computing the EulerPoincaré characteristics of sign conditions
 Comput. Complexity
"... Abstract. Computing various topological invariants of semialgebraic 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 semialgebraic sets in single exponential time etc. However, an alg ..."
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Cited by 7 (5 self)
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Abstract. Computing various topological invariants of semialgebraic 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 semialgebraic sets in single exponential time etc. However, an algorithm for computing all the Betti numbers of a given semialgebraic 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 semialgebraic set can be computed with the same complexity. The best previously known single exponential time algorithm for computing the Euler–Poincaré characteristic of semialgebraic sets worked only for a more restricted class of closed semialgebraic sets and had a complexity of (ksd) O(k).
Betti numbers of semialgebraic sets defined by partly quadratic systems of polynomials
, 2007
"... ... degX (P) ≤ d, P ∈ P, #(P) = s, and S ⊂ Rℓ+k a semialgebraic set defined by 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] an ..."
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Cited by 6 (2 self)
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... degX (P) ≤ d, P ∈ P, #(P) = s, and S ⊂ Rℓ+k a semialgebraic set defined by 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 semialgebraic sets defined by polynomials of degree d and 2, respectively. We also describe algorithms for computing the EulerPoincaré 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)
REFINED BOUNDS ON THE NUMBER OF CONNECTED COMPONENTS OF SIGN CONDITIONS ON A VARIETY
"... Abstract. Let R be a real closed field, P, Q ⊂ R[X1,..., Xk] finite subsets of polynomials, with the degrees of the polynomials in P (resp. Q) bounded by d (resp. d0). Let V ⊂ R k be the real algebraic variety defined by the polynomials in Q and suppose that the real dimension of V is bounded by k ′ ..."
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Abstract. Let R be a real closed field, P, Q ⊂ R[X1,..., Xk] finite subsets of polynomials, with the degrees of the polynomials in P (resp. Q) bounded by d (resp. d0). Let V ⊂ R k be the real algebraic variety defined by the polynomials in Q and suppose that the real dimension of V is bounded by k ′. We prove that the number of semialgebraically connected components of the realizations of all realizable sign conditions of the family P on V is bounded by k ′ X 4 j=0 j “ s + 1 Fd,d0,k,k j ′(j), where s = card P, and Fd,d0,k,k ′(j) = ` k+1 k−k ′ ´ (2d0) +j+1 k−k ′ dj max{2d0, d} k ′ −j + 2(k − j + 1). In case 2d0 ≤ d, the above bound can be written simply as k ′
Algorithmic Semialgebraic Geometry and Topology – Recent Progress and Open Problems (expository article, 73 pages), to appear
 in AMS Contemporary Mathematics Series, Proceedings the Summer Research Conference on Discrete and Computational Geometry – Twenty years later, Snowbird
, 2006
"... Abstract. In this lecture we introduce semialgebraic sets, TarskiSeidenberg principle, give basic definitions of homology and cohomology groups of semialgebraic sets, and state certain quantitative results which give tight bounds on the ranks of these groups. We also state several ..."
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Cited by 3 (1 self)
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Abstract. In this lecture we introduce semialgebraic sets, TarskiSeidenberg principle, give basic definitions of homology and cohomology groups of semialgebraic sets, and state certain quantitative results which give tight bounds on the ranks of these groups. We also state several
An asymptotically tight bound on the number of connected components of realizable sign conditions
"... 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 ..."
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Cited by 3 (1 self)
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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
Bounding the Betti numbers and computing the EulerPoincaré characteristic of semialgebraic sets defined by partly quadratic systems of polynomials
"... deg X (P) ≤ d, P ∈ P,#(P) = s, and S ⊂ R ℓ+k a semialgebraic set defined by 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 ℓ 2 (O(s + ℓ + m)ℓd) k+2m. This is a common generalization of previous results ..."
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Cited by 3 (3 self)
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deg X (P) ≤ d, P ∈ P,#(P) = s, and S ⊂ R ℓ+k a semialgebraic set defined by 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 ℓ 2 (O(s + ℓ + m)ℓd) k+2m. This is a common generalization of previous results in [4] and [3] on bounding the Betti numbers of closed semialgebraic sets defined by polynomials of degree d and 2, respectively. We also describe an algorithm for computing the EulerPoincaré characteristic of such sets, generalizing similar algorithms described in [4, 12]. The complexity of the algorithm is bounded by (ℓsmd) O(m(m+k)).
Incidence theorems for pseudoflats
"... We prove PachSharir 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 ..."
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We prove PachSharir 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