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12
Variations by complexity theorists on three themes of
 Computational Complexity
, 2005
"... This paper surveys some connections between geometry and complexity. A main role is played by some quantities —degree, Euler characteristic, Betti numbers — associated to algebraic or semialgebraic sets. This role is twofold. On the one hand, lower bounds on the deterministic time (sequential and pa ..."
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Cited by 12 (4 self)
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This paper surveys some connections between geometry and complexity. A main role is played by some quantities —degree, Euler characteristic, Betti numbers — associated to algebraic or semialgebraic sets. This role is twofold. On the one hand, lower bounds on the deterministic time (sequential and parallel) necessary to decide a set S are established as functions of these quantities associated to S. The optimality of some algorithms is obtained as a consequence. On the other hand, the computation of these quantities gives rise to problems which turn out to be hard (or complete) in different complexity classes. These two kind of results thus turn the quantities above into measures of complexity in two quite different ways. 1
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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Cited by 10 (3 self)
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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 8 (5 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.
The Number of Embeddings of Minimally Rigid Graphs
 GEOMETRY © 2003 SPRINGERVERLAG NEW YORK INC.
, 2003
"... Rigid frameworks in some Euclidean space are embedded graphs having a unique local realization (up to Euclidean motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with n vertices. We show that, mo ..."
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Cited by 7 (2 self)
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Rigid frameworks in some Euclidean space are embedded graphs having a unique local realization (up to Euclidean motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with n vertices. We show that, modulo planar rigid motions, this number is at most � � 2n−4 n ≈ 4. We also exhibit several families which realize lower bounds n−2 of the order of 2n,2.21n and 2.28n. For the upper bound we use techniques from complex algebraic geometry, based on the (projective) Cayley–Menger variety CM2,n (C) ⊂ P n ( 2)−1(C) over the complex numbers C. In this context, point configurations are represented by coordinates given by squared distances between all pairs of points. Sectioning the variety with 2n − 4 hyperplanes yields at most deg(CM2,n) zerodimensional components, and one finds this degree to be D2,n � � = 1 2n−4. The lower bounds are related to inductive constructions of minimally rigid graphs 2 n−2 via Henneberg sequences. The same approach works in higher dimensions. In particular, we show that it leads to an upper bound of 2D3,n = (2n−3 /(n − 2)) � � 2n−6 for the number of spatial embeddings n−3 with generic edge lengths of the 1skeleton of a simplicial polyhedron, up to rigid motions. Our technique can also be adapted to the nonEuclidean case.
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
Formal proof, computation, and the construction problem in algebraic geometry
"... It has become a classical technique to turn to theoretical computer science to provide computational tools for algebraic geometry. A more recent transformation is that now we also get logical tools, and these too should be useful in the study of algebraic varieties. The purpose of this note is to co ..."
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It has become a classical technique to turn to theoretical computer science to provide computational tools for algebraic geometry. A more recent transformation is that now we also get logical tools, and these too should be useful in the study of algebraic varieties. The purpose of this note is to consider a very small part of this picture, and try to motivate the study of computer theoremproving techniques by looking at how they might be relevant to a particular class of problems in algebraic geometry. This is only an informal discussion, based more on questions and possible research directions than on actual results. This note amplifies the themes discussed in my talk at the “Arithmetic and Differential Galois Groups ” conference (March 2004, Luminy), although many specific points in the discussion were only finished more recently. I would like to thank: André Hirschowitz and Marco Maggesi, for their invaluable insights about computerformalized mathematics as it relates
On a topological fractional Helly theorem
, 2005
"... We prove a new fractional Helly theorem for families of sets obeying topological conditions. More precisely, we show that the nerve of a finite family of open sets (and of subcomplexes of cell complexes) in R d is kLeray where k depends on the dimension d and the homological intersection complexity ..."
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We prove a new fractional Helly theorem for families of sets obeying topological conditions. More precisely, we show that the nerve of a finite family of open sets (and of subcomplexes of cell complexes) in R d is kLeray where k depends on the dimension d and the homological intersection complexity of the family. This implies fractional Helly number k + 1 for families F: For every α> 0 there is a β(α)> 0 such that for sets F1, F2,..., Fn � ∈ F with i∈I Fi for at least ⌊α � � n ⌋ sets I ⊆ {1, 2,..., n} of size k + 1, there exists k+1 a point which is common to at least ⌊βn ⌋ of the Fi. Moreover, we obtain a topological (p, q)theorem. Our result contains the (p,q)theorem for good covers of Alon, Kalai, Matouˇsek, and Meshulam [2] as a special case. The proof uses a spectral sequence argument. The same method is then used to reprove a homological version of a nerve theorem of Björner. 1
Contemporary Mathematics Algorithmic Semialgebraic Geometry and Topology – Recent Progress and Open Problems
"... Abstract. We give a survey of algorithms for computing topological invariants of semialgebraic sets with special emphasis on the more recent developments in designing algorithms for computing the Betti numbers of semialgebraic sets. Aside from describing these results, we discuss briefly the backgr ..."
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Abstract. We give a survey of algorithms for computing topological invariants of semialgebraic sets with special emphasis on the more recent developments in designing algorithms for computing the Betti numbers of semialgebraic sets. Aside from describing these results, we discuss briefly the background as well as the importance of these problems, and also describe the main tools from algorithmic semialgebraic geometry, as well as algebraic topology, which make these advances possible. We end with a list of open problems.