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56
A General Approach To Removing Degeneracies
 SIAM J. Computing
, 1991
"... We wish to increase the power of an arbitrary algorithm designed for nondegenerate input, by allowing it to execute on all inputs. We concentrate on infinitesimal symbolic perturbations that do not affect the output for inputs in general position. Otherwise, if the problem mapping is continuous, th ..."
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Cited by 54 (6 self)
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We wish to increase the power of an arbitrary algorithm designed for nondegenerate input, by allowing it to execute on all inputs. We concentrate on infinitesimal symbolic perturbations that do not affect the output for inputs in general position. Otherwise, if the problem mapping is continuous, the input and output space topology are at least as coarse as the real euclidean one and the output space is connected, then our perturbations make the algorithm produce an output arbitrarily close or identical to the correct one. For a special class of algorithms, which includes several important algorithms in computational geometry,we describe a deterministic method that requires no symbolic computation. Ignoring polylogarithmic factors, this method increases only the worstcase bit complexity by a multiplicative factor which is linear in the dimension of the geometric space. For general algorithms, a randomized scheme with arbitrarily high probability of success is proposed; the bit complexity is then bounded by a smalldegree polynomial in the original worstcase complexity. In addition to being simpler than previous ones, these are the first efficient perturbation methods.
Quantifier Elimination for Real Algebra  the Quadratic Case and Beyond
 AAECC
, 1993
"... . We present a new, "elementary" quantifier elimination method for various special cases of the general quantifier elimination problem for the firstorder theory of real numbers. These include the elimination of one existential quantifier 9x in front of quantifierfree formulas restricte ..."
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Cited by 44 (4 self)
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. We present a new, "elementary" quantifier elimination method for various special cases of the general quantifier elimination problem for the firstorder theory of real numbers. These include the elimination of one existential quantifier 9x in front of quantifierfree formulas restricted by a nontrivial quadratic equation in x (the case considered also in [7]), and more generally in front of arbitrary quantifierfree formulas involving only polynomials that are quadratic in x. The method generalizes the linear quantifier elimination method by virtual substitution of test terms in [9]. It yields a quantifier elimination method for an arbitrary number of quantifiers in certain formulas involving only linear and quadratic occurences of the quantified variables. Moreover, for existential formulas ' of this kind it yields sample answers to the query represented by '. The method is implemented in reduce as part of the redlog package (see [4, 5]). Experiments show that the method is appl...
An Algorithm for Sums of Squares of Real Polynomials
 Journal of Pure and Applied Algebra
"... This paper was written while the rst author was a visitor at Dortmund University. She gratefully thanks the Deutscher Akademischer Austauschdienst for funding for this visit, as well as Professor E. Becker and his assistants for their warm hospitality during her stay. AN ALGORITHM FOR SUMS OF SQUAR ..."
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Cited by 33 (0 self)
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This paper was written while the rst author was a visitor at Dortmund University. She gratefully thanks the Deutscher Akademischer Austauschdienst for funding for this visit, as well as Professor E. Becker and his assistants for their warm hospitality during her stay. AN ALGORITHM FOR SUMS OF SQUARES OF REAL POLYNOMIALS 5
Diffractive Nonlinear Geometric Optics With Rectification
 Indiana Univ. Math. J
, 1998
"... . This paper studies high frequency solutions of nonlinear hyperbolic equations for time scales at which diffractive effects and nonlinear effects are both present in the leading term of approximate solutions. The key innovation is the analysis of rectification effects, that is the interaction of t ..."
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Cited by 32 (4 self)
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. This paper studies high frequency solutions of nonlinear hyperbolic equations for time scales at which diffractive effects and nonlinear effects are both present in the leading term of approximate solutions. The key innovation is the analysis of rectification effects, that is the interaction of the nonoscillatory local mean field with the rapidly oscillating fields. The main results prove that in the limit of frequency tending to infinity, the relative error in our approximate solutions tends to zero. One of our main conclusions is that for oscillatory fields associated with wave vectors on curved parts of the characteristic variety, the interaction is negligible to leading order. For wave vectors on flat parts of the variety, the interaction is spelled out in detail. Outline. x1. Introduction. x2. The ansatz and the first profile equations. x3. Large time asymptotics for linear symmetric hyperbolic systems. x4. Profile equations, continuation. x5. Solvability of the profile eq...
Generalized polar varieties: Geometry and algorithms
, 2004
"... Let V be a closed algebraic subvariety of the n–dimensional projective space over the complex or real numbers and suppose that V is non–empty and equidimensional. The classic notion of a polar variety of V associated with a given linear subvariety of the ambient space of V was generalized and motiva ..."
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Cited by 26 (8 self)
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Let V be a closed algebraic subvariety of the n–dimensional projective space over the complex or real numbers and suppose that V is non–empty and equidimensional. The classic notion of a polar variety of V associated with a given linear subvariety of the ambient space of V was generalized and motivated in [2]. As particular instances of this notion of a generalized polar variety one reobtains the classic one and an alternative type of a polar varietiy, called dual. As main result of the present paper we show that for a generic choice of their parameters the generalized polar varieties of V are empty or equidimensional and smooth in any regular point of V. In the case that the variety V is affine and smooth and has a complete intersection ideal of definition, we are able, for a generic parameter choice, to describe locally the generalized polar varieties of V by explicit equations. Finally, we indicate how this description may be used in order to design in
Variations by complexity theorists on three themes of Euler, . . .
 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 11 (3 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.
An orientation for the SU(2)representation space of knot groups
 Topology Appl
, 2003
"... In 1985 Casson constructed a new integer valued invariant for homology 3–spheres (see [AM90, GM92]). His construction is based on properties of SU(2)–representation spaces. A surprising and important corollary is that a knot k ⊂ S3 has Property P if ∆′′k(1) 6 = 0 where ∆k(t) is the normalized ..."
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Cited by 10 (1 self)
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In 1985 Casson constructed a new integer valued invariant for homology 3–spheres (see [AM90, GM92]). His construction is based on properties of SU(2)–representation spaces. A surprising and important corollary is that a knot k ⊂ S3 has Property P if ∆′′k(1) 6 = 0 where ∆k(t) is the normalized
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 10 (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.
An effective version of Pólya's theorem on positive definite forms
, 1995
"... Given a real homogeneous polynomial F , strictly positive in the nonnegative orthant, P'olya's theorem says that for a sufficiently large exponent p the coefficients of F (x1 ; . . . ; xn ) \Delta (x1 + \Delta \Delta \Delta +xn ) p are strictly positive. The smallest such p will be call ..."
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Cited by 10 (0 self)
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Given a real homogeneous polynomial F , strictly positive in the nonnegative orthant, P'olya's theorem says that for a sufficiently large exponent p the coefficients of F (x1 ; . . . ; xn ) \Delta (x1 + \Delta \Delta \Delta +xn ) p are strictly positive. The smallest such p will be called the P'olya exponent of F . We present a new proof for P'olya's result, which allows us to obtain an explicit upper bound on the P'olya exponent when F has rational coefficients. An algorithm to obtain reasonably good bounds for specific instances is also derived. P'olya's theorem has appeared before in constructive solutions of Hilbert's 17th problem for positive definite forms [4]. We also present a different procedure to do this kind of construction. 1 Introduction In 1928 G. P'olya [7] proved the following theorem (see also [5]): Theorem 1.1 (P'olya) Let F (x 1 ; . . . ; x n ) be a real homogeneous polynomial which is positive in x i 0, P x i ? 0. Then, for a sufficiently large integer p, ...