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Different bounds on the different Betti numbers of semialgebraic sets
 Proceedings of the ACM Symposium on Computational Geometry
, 2001
"... A classic result in real algebraic geometry due to OleinikPetrovsky, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semialgebraic sets. This bound is tight as one can construct examples having that many connected components. However, till now no signif ..."
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Cited by 18 (8 self)
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A classic result in real algebraic geometry due to OleinikPetrovsky, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semialgebraic sets. This bound is tight as one can construct examples having that many connected components. However, till now
On Bounding the Betti Numbers and Computing the Euler Characteristic of Semialgebraic Sets
, 1996
"... In this paper we give a new bound on the sum of the Betti numbers of semialgebraic sets. This extends a wellknown bound due to Oleinik and Petrovsky [19], Thom [23] and Milnor [18]. In separate papers they proved that the sum of the Betti numbers of a semialgebraic set S ae R k ; defined by P ..."
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Cited by 58 (22 self)
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In this paper we give a new bound on the sum of the Betti numbers of semialgebraic sets. This extends a wellknown bound due to Oleinik and Petrovsky [19], Thom [23] and Milnor [18]. In separate papers they proved that the sum of the Betti numbers of a semialgebraic set S ae R k ; defined by P
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 7 (3 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
Computing the first Betti number and describing the connected components of semialgebraic sets
 In Proc. STOC
, 2005
"... Abstract. In this paper we describe a singly exponential algorithm for computing the first Betti number of a given semialgebraic set. Singly exponential algorithms for computing the zeroth Betti number, and the EulerPoincaré characteristic, were known before. No singly exponential algorithm was k ..."
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Cited by 1 (0 self)
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Abstract. In this paper we describe a singly exponential algorithm for computing the first Betti number of a given semialgebraic set. Singly exponential algorithms for computing the zeroth Betti number, and the EulerPoincaré characteristic, were known before. No singly exponential algorithm
New bounds on the Betti numbers of semialgebraic sets and arrangements of real algebraic hypersurfaces
"... A classic result in real algebraic geometry due to OleinikPetrovsky, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semialgebraic sets. This bound is tight as one can construct examples having that many connected components. However, till now no sign ..."
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Cited by 8 (1 self)
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no significantly better bounds were known on the individual higher Betti numbers. In this paper we prove separate bounds on the different Betti numbers of basic semialgebraic sets, as well as arrangements of algebraic hypersurfaces. These are the first results in this direction. 1 Introduction
Algorithms in Semialgebraic Geometry
, 1996
"... In this thesis we present new algorithms to solve several very general problems of semialgebraic geometry. These are currently the best algorithms for solving these problems. In addition, we have proved new bounds on the topological complexity of real semialgebraic sets, in terms of the paramete ..."
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Cited by 9 (0 self)
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In this thesis we present new algorithms to solve several very general problems of semialgebraic geometry. These are currently the best algorithms for solving these problems. In addition, we have proved new bounds on the topological complexity of real semialgebraic sets, in terms
How much should we trust differencesindifferences estimates? Quarterly Journal of Economics 119:249–75
, 2004
"... Most papers that employ DifferencesinDifferences estimation (DD) use many years of data and focus on serially correlated outcomes but ignore that the resulting standard errors are inconsistent. To illustrate the severity of this issue, we randomly generate placebo laws in statelevel data on fema ..."
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Cited by 775 (1 self)
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Most papers that employ DifferencesinDifferences estimation (DD) use many years of data and focus on serially correlated outcomes but ignore that the resulting standard errors are inconsistent. To illustrate the severity of this issue, we randomly generate placebo laws in statelevel data
Results 1  10
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1,696,143