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222
Revisiting Two Theorems of Curto and Fialkow on Moment Matrices
, 2004
"... We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence... ..."
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Cited by 18 (4 self)
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We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence...
Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets (Extended Abstract)
 J. COMPL
, 2004
"... We define counting #P classes #P ¡ and in the BlumShubSmale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over ¢ , or of systems of polynomial equalities over £ , respectively, turn ou ..."
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Cited by 18 (11 self)
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We define counting #P classes #P ¡ and in the BlumShubSmale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over ¢ , or of systems of polynomial equalities over £ , respectively, turn out to be natural complete problems in these classes. We investigate to what extent the new counting classes capture the complexity of computing basic topological invariants of semialgebraic sets (over ¢ ) and algebraic sets (over £). We prove that the problem to compute the (modified) Euler characteristic of semialgebraic sets is FP #P¤complete, and that the problem to compute the geometric degree of complex algebraic sets is FP #P¥complete. We also define new counting complexity classes GCR and GCC in the classical Turing model via taking Boolean parts of the classes above, and show that the problems to compute the Euler characteristic and the geometric degree of (semi)algebraic sets given by integer polynomials are complete in these classes. We complement the results in the Turing model by proving, for all k ¦ ∈ , the FPSPACEhardness of the problem of computing the kth Betti number of the set of real zeros of a given integer polynomial. This holds with respect to the singular homology as well as for the BorelMoore homology.
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 16 (7 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 no significantly better bounds were known on the individual higher Betti numbers. We prove...
On the complexity of real functions
, 2005
"... We establish a new connection between the two most common traditions in the theory of real computation, the BlumShubSmale model and the Computable Analysis approach. We then use the connection to develop a notion of computability and complexity of functions over the reals that can be viewed as an ..."
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Cited by 15 (5 self)
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We establish a new connection between the two most common traditions in the theory of real computation, the BlumShubSmale model and the Computable Analysis approach. We then use the connection to develop a notion of computability and complexity of functions over the reals that can be viewed as an extension of both models. We argue that this notion is very natural when one tries to determine just how “difficult ” a certain function is for a very rich class of functions. 1
Triangular Decomposition of SemiAlgebraic Systems
, 2010
"... Regular chains and triangular decompositions are fundamental and welldeveloped tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue: semialgebraic systems. We show that any such system can be dec ..."
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Cited by 14 (9 self)
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Regular chains and triangular decompositions are fundamental and welldeveloped tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue: semialgebraic systems. We show that any such system can be decomposed into finitely many regular semialgebraic systems. We propose two specifications of such a decomposition and present corresponding algorithms. Under some assumptions, one type of decomposition can be computed in singly exponential time w.r.t. the number of variables. We implement our algorithms and the experimental results illustrate their effectiveness.
Positive polynomials in scalar and matrix variables, the spectral theorem, and optimization
 , in vol. Structured Matrices and Dilations. A Volume Dedicated to the Memory of Tiberiu Constantinescu
"... We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second par ..."
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Cited by 13 (3 self)
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We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second part of the survey focuses on recently discovered connections between real algebraic geometry and optimization as well as polynomials in matrix variables and some control theory problems. These new applications have prompted a series of recent studies devoted to the structure of positivity and convexity in a free ∗algebra, the appropriate setting for analyzing inequalities on polynomials having matrix variables. We sketch some of these developments, add to them and comment on the rapidly growing literature.
Testing sign conditions on a multivariate polynomial and applications
 MATHEMATICS IN COMPUTER SCIENCE
"... Let f be a polynomial in Q[X1,..., Xn] of degree D. We focus on testing the emptiness and computing at least one point in each connected component of the semialgebraic set defined by f> 0 (or f < 0 or f = 0). To this end, the problem is reduced to computing at least one point in each connected c ..."
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Cited by 13 (6 self)
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Let f be a polynomial in Q[X1,..., Xn] of degree D. We focus on testing the emptiness and computing at least one point in each connected component of the semialgebraic set defined by f> 0 (or f < 0 or f = 0). To this end, the problem is reduced to computing at least one point in each connected component of a hypersurface defined by f − e = 0 for e ∈ Q positive and small enough. We provide an algorithm allowing us to determine a positive rational number e which is small enough in this sense. This is based on the efficient computation of the set of generalized critical values of the mapping f: y ∈ C n → f(y) ∈ C which is the union of the classical set K0(f) of critical values of the mapping f and K∞(f) of asymptotic critical values of the mapping f. Then, we show how to use the computation of generalized critical values in order to obtain an efficient algorithm deciding the emptiness of a semialgebraic set defined by a single inequality or a single inequation. At last, we show how to apply our contribution to determining if a hypersurface contains real regular points. We provide complexity estimates for probabilistic versions of the latter algorithms which are within O(n 7 D 4n) arithmetic operations in Q. The paper ends with practical experiments showing the efficiency of our approach.
Reasoning about probabilistic sequential programs ∗
"... A complete and decidable Hoarestyle calculus for iterationfree probabilistic sequential programs is presented using a state logic with truthfunctional propositional (not arithmetical) connectives. 1 ..."
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Cited by 12 (11 self)
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A complete and decidable Hoarestyle calculus for iterationfree probabilistic sequential programs is presented using a state logic with truthfunctional propositional (not arithmetical) connectives. 1
On the frontier of polynomial computations in tropical geometry
 Journal of Symbolic Computation
"... Abstract. We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is connected, as well as counting the number of connected components. We cha ..."
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Cited by 12 (0 self)
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Abstract. We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is connected, as well as counting the number of connected components. We characterize the borderline between tractable and hard computations by proving N Phardness and #Phardness results under various strong restrictions of the input data, as well as providing polynomial time algorithms for various other restrictions. 1.
Polynomial Systems with Few Real Zeroes
"... Abstract. We study some systems of polynomials whose support lies in the convex hull of a circuit, giving an upper bound for their numbers of real solutions which is sharp in some instances. This upper bound is nontrivial in that it is smaller than either the Kouchnirenko or the Khovanskii bounds f ..."
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Cited by 12 (10 self)
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Abstract. We study some systems of polynomials whose support lies in the convex hull of a circuit, giving an upper bound for their numbers of real solutions which is sharp in some instances. This upper bound is nontrivial in that it is smaller than either the Kouchnirenko or the Khovanskii bounds for these systems. When the support is exactly a circuit whose affine span is Z n, this bound is 2n + 1, while the Khovanskii bound is exponential in n 2. The bound 2n + 1 is sharp and can be attained only for nondegenerate circuits. Our methods are based on computing an eliminant and involve a mixture of combinatorics, geometry, and arithmetic.