Results 1  10
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1,497,505
Computing the first few Betti numbers of semialgebraic sets in single exponential time
, 2006
"... In this paper we describe an algorithm that takes as input a description of a semialgebraic set S ⊂ Rk, defined by a Boolean formula with atoms of the form P> 0,P < 0,P = 0 for P ∈ P ⊂ R[X1,...,Xk], and outputs the first ℓ + 1 Betti numbers of S, b0(S),...,bℓ(S). The complexity of the algorith ..."
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Cited by 28 (12 self)
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of the algorithm is (sd) kO(ℓ), where where s = #(P) and d = maxP ∈P deg(P), which is singly exponential in k for ℓ any fixed constant. Previously, singly exponential time algorithms were known only for computing the EulerPoincaré characteristic, the zeroth and the first Betti numbers.
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
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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] 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
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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. Moreover, we prove some purely mathematical theorems on the number of connected components and on the existence of small rational points in a given semialgebraic set. The second part of this thesis deals with connectivity questions of semialgebraic sets. We develop new techniques in order to give a...
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 projections of semialgebraic sets defined by few quadratic inequalities
 in Discrete and Computational Geometry, available at [arXiv:math.AG/0602398]. SEMIALGEBRAIC GEOMETRY AND TOPOLOGY 73
"... Abstract. Let S ⊂ R k+m be a compact semialgebraic set defined by P1 ≥ 0,..., Pℓ ≥ 0, where Pi ∈ R[X1,..., Xk, Y1,..., Ym], and deg(Pi) ≤ 2, 1 ≤ i ≤ ℓ. Let π denote the standard projection from R k+m onto R m. We prove that for any q> 0, the sum of the first q Betti numbers of π(S) is bounded b ..."
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Cited by 9 (7 self)
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Abstract. Let S ⊂ R k+m be a compact semialgebraic set defined by P1 ≥ 0,..., Pℓ ≥ 0, where Pi ∈ R[X1,..., Xk, Y1,..., Ym], and deg(Pi) ≤ 2, 1 ≤ i ≤ ℓ. Let π denote the standard projection from R k+m onto R m. We prove that for any q> 0, the sum of the first q Betti numbers of π(S) is bounded
Computing the Betti numbers of semialgebraic sets defined by partly quadratic systems of polynomials
 2009) ZBL 1171.14040 MR 2501518
"... ... with degX (P) ≤ d, P ∈ P, #(P) = s. Let S ⊂ Rℓ+k be a semialgebraic set defined by a Boolean formula without negations, with atoms P = 0, P ≥ 0, P ≤ 0, P ∈ P ∪ Q. We describe an algorithm for computing the the Betti numbers of S generalizing a similar algorithm described in [6]. The complexit ..."
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Cited by 9 (6 self)
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... with degX (P) ≤ d, P ∈ P, #(P) = s. Let S ⊂ Rℓ+k be a semialgebraic set defined by a Boolean formula without negations, with atoms P = 0, P ≥ 0, P ≤ 0, P ∈ P ∪ Q. We describe an algorithm for computing the the Betti numbers of S generalizing a similar algorithm described in [6
BOUNDING THE EQUIVARIANT BETTI NUMBERS AND COMPUTING THE GENERALIZED EULERPOINCARÉ CHARACTERISTIC OF SYMMETRIC SEMIALGEBRAIC SETS
, 2014
"... Let R be a real closed field. The problem of obtaining tight bounds on the Betti numbers of semialgebraic subsets of R k in terms of the number and degrees of the defining polynomials has been an important problem in real algebraic geometry with the first results due to Oleĭnik and Petrovskiĭ, Tho ..."
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Cited by 3 (3 self)
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on the ordinary Betti numbers of the projection of a compact semialgebraic set improving for any fixed degree the best previously known bound for this problem due to Gabrielov, Vorobjov and Zell. As another application of our methods we obtain polynomial time (for fixed degrees) algorithms for computing
Computing the top Betti numbers of semialgebraic sets defined by quadratic inequalities in polynomial time
, 2006
"... For any ℓ> 0, we present an algorithm which takes as input a semialgebraic set, S, defined by P1 ≤ 0,..., Ps ≤ 0, where each Pi ∈ R[X1,..., Xk] has degree ≤ 2, and computes the top ℓ Betti numbers of S, bk−1(S),..., bk−ℓ(S), in polynomial time. The complexity of the algorithm, stated more prec ..."
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Cited by 3 (1 self)
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For any ℓ> 0, we present an algorithm which takes as input a semialgebraic set, S, defined by P1 ≤ 0,..., Ps ≤ 0, where each Pi ∈ R[X1,..., Xk] has degree ≤ 2, and computes the top ℓ Betti numbers of S, bk−1(S),..., bk−ℓ(S), in polynomial time. The complexity of the algorithm, stated more
Bounding the Betti numbers and computing the EulerPoincaré characteristic of semialgebraic sets defined by partly quadratic systems of polynomials
, 2008
"... ... 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 resu ..."
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Cited by 8 (5 self)
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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
Results 1  10
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1,497,505