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168
On the Combinatorial and Algebraic Complexity of Quantifier Elimination
, 1996
"... In this paper, a new algorithm for performing quantifier elimination from first order formulas over real closed fields is given. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this date. A new feature of this algorithm is that the role of th ..."
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Cited by 200 (28 self)
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In this paper, a new algorithm for performing quantifier elimination from first order formulas over real closed fields is given. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this date. A new feature of this algorithm is that the role of the algebraic part (the dependence on the degrees of the input polynomials) and the combinatorial part (the dependence on the number of polynomials) are separated. Another new feature is that the degrees of the polynomials in the equivalent quantifierfree formula that is output, are independent of the number of input polynomials. As special cases of this algorithm, new and improved algorithms for deciding a sentence in the first order theory over real closed fields, and also for solving the existential problem in the first order theory over real closed fields, are obtained.
Some Concrete Aspects Of Hilbert's 17th Problem
 In Contemporary Mathematics
, 1996
"... This paper is dedicated to the memory of Raphael M. Robinson and Olga TausskyTodd. 1. Introduction ..."
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Cited by 99 (4 self)
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This paper is dedicated to the memory of Raphael M. Robinson and Olga TausskyTodd. 1. Introduction
Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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Cited by 81 (20 self)
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
Sums of squares, moment matrices and optimization over polynomials
, 2008
"... We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NPhard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory ..."
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Cited by 63 (8 self)
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We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NPhard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory of sums of squares of polynomials. We present these hierarchies of approximations and their main properties: asymptotic/finite convergence, optimality certificate, and extraction of global optimum solutions. We review the mathematical tools underlying these properties, in particular, some sums of squares representation results for positive polynomials, some results about moment matrices (in particular, of Curto and Fialkow), and the algebraic eigenvalue method for solving zerodimensional systems of polynomial equations. We try whenever possible to provide detailed proofs and background.
Random Worlds and Maximum Entropy
 In Proc. 7th IEEE Symp. on Logic in Computer Science
, 1994
"... Given a knowledge base KB containing firstorder and statistical facts, we consider a principled method, called the randomworlds method, for computing a degree of belief that some formula ' holds given KB . If we are reasoning about a world or system consisting of N individuals, then we can co ..."
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Cited by 49 (12 self)
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Given a knowledge base KB containing firstorder and statistical facts, we consider a principled method, called the randomworlds method, for computing a degree of belief that some formula ' holds given KB . If we are reasoning about a world or system consisting of N individuals, then we can consider all possible worlds, or firstorder models, with domain f1; : : : ; Ng that satisfy KB , and compute the fraction of them in which ' is true. We define the degree of belief to be the asymptotic value of this fraction as N grows large. We show that when the vocabulary underlying ' and KB uses constants and unary predicates only, we can naturally associate an entropy with each world. As N grows larger, there are many more worlds with higher entropy. Therefore, we can use a maximumentropy computation to compute the degree of belief. This result is in a similar spirit to previous work in physics and artificial intelligence, but is far more general. Of equal interest to the result itself are...
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 49 (18 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 1 0; : : : ; P s 0; deg(P i ) d; 1 i s; is bounded by (O(sd)) k : Given a semialgebraic set S ae R k defined as the intersection of a real variety, Q = 0; deg(Q) d; whose real dimension is k 0 ; with a set defined by a quantifierfree Boolean formula with atoms of the form, P i = 0; P i ? 0; P i ! 0; deg(P i ) d; 1 i s; we prove that the sum of the Betti numbers of S is bounded by s k 0 (O(d)) k : In the special case, when S is defined by Q = 0; P 1 ? 0; : : : ; P s ? 0; we have a slightly tighter bound of \Gamma s k 0 \Delta (O(d)) k : This result generalises the OleinikPetrovskyThomMilnor bound in two directions. Firstly, our bound applies to arbitrary semialg...
Sums of squares of regular functions on real algebraic varieties
 Tran. Amer. Math. Soc
, 1999
"... Abstract. Let V be an affine algebraic variety over R (or any other real closed field R). We ask when it is true that every positive semidefinite (psd) polynomial function on V is a sum of squares (sos). We show that for dim V ≥ 3 the answer is always negative if V has a real point. Also, if V is a ..."
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Cited by 48 (10 self)
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Abstract. Let V be an affine algebraic variety over R (or any other real closed field R). We ask when it is true that every positive semidefinite (psd) polynomial function on V is a sum of squares (sos). We show that for dim V ≥ 3 the answer is always negative if V has a real point. Also, if V is a smooth nonrational curve all of whose points at infinity are real, the answer is again negative. The same holds if V is a smooth surface with only real divisors at infinity. The “compact ” case is harder. We completely settle the case of smooth curves of genus ≤ 1: If such a curve has a complex point at infinity, then every psd function is sos, provided the field R is archimedean. If R is not archimedean, there are counterexamples of genus 1.
Computing roadmaps of semialgebraic sets on a variety
 Journal of the AMS
, 1997
"... Let R be a real closed field, Z(Q) a real algebraic variety defined as the zero set of a polynomial Q ∈ R[X1,...,Xk]andSasemialgebraic subset of Z(Q), defined by a Boolean formula with atoms of the form P < 0,P> 0,P =0withP∈P, where P is a finite subset of R[X1,...,Xk]. A semialgebraic set C ..."
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Cited by 42 (14 self)
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Let R be a real closed field, Z(Q) a real algebraic variety defined as the zero set of a polynomial Q ∈ R[X1,...,Xk]andSasemialgebraic subset of Z(Q), defined by a Boolean formula with atoms of the form P < 0,P> 0,P =0withP∈P, where P is a finite subset of R[X1,...,Xk]. A semialgebraic set C is semialgebraically connected if it is nonempty and is not
A New Algorithm to Find a Point in Every Cell Defined by a Family of Polynomials
 B. Caviness and J. Johnson Eds., SpringerVerlag
, 1995
"... We consider s polynomials P 1 ; : : : ; P s in k ! s variables with coefficients in an ordered domain A contained in a real closed field R, each of degree at most d. We present a new algorithm which computes a point in each connected component of each nonempty sign condition over P 1 ; : : : ; P s ..."
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Cited by 39 (8 self)
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We consider s polynomials P 1 ; : : : ; P s in k ! s variables with coefficients in an ordered domain A contained in a real closed field R, each of degree at most d. We present a new algorithm which computes a point in each connected component of each nonempty sign condition over P 1 ; : : : ; P s . The output is the set of points together with the sign condition at each point. The algorithm uses s(s=k) k d O(k) arithmetic operations in A. The algorithm is nearly optimal in the sense that the size of the output can be as large as s(O(sd=k)) k . Previous algorithms of Canny and Renegar used (sd) O(k) operations [5, 7, 8, 15]. We use either these algorithms in the case s = 1 as a subroutine in our algorithm. As a bonus, our algorithm yields an independent proof of the bound on the number of connected components in all nonempty sign conditions ([14]) and also yields an independent proof of a theorem of Warren 1 Courant Institute of Mathematical Sciences, New York University, N...