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52
Semidefinite Programming Relaxations for Semialgebraic Problems
, 2001
"... A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions reducible to a finite number of polynomial equalities and inequalities, it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility. The mai ..."
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Cited by 220 (19 self)
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A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions reducible to a finite number of polynomial equalities and inequalities, it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility. The main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials, and some results from real algebraic geometry. The techniques provide a constructive approach for finding bounded degree solutions to the Positivstellensatz, and are illustrated with examples from diverse application fields.
Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization
, 2000
"... ..."
Mathematical Problems for the Next Century
 Mathematical Intelligencer
, 1998
"... This report is my response. ..."
Lower bounds on Hilbert's Nullstellensatz and propositional proofs
 PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
, 1996
"... The socalled weak form of Hilbert's Nullstellensatz says that a system of algebraic equations over a field, Qj(x) = 0, does not have a solution in the algebraic closure if and only if 1 is in the ideal generated by the polynomials (?,(*) • We shall prove a lower bound on the degrees of polynomials ..."
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Cited by 61 (18 self)
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The socalled weak form of Hilbert's Nullstellensatz says that a system of algebraic equations over a field, Qj(x) = 0, does not have a solution in the algebraic closure if and only if 1 is in the ideal generated by the polynomials (?,(*) • We shall prove a lower bound on the degrees of polynomials P,(x) such that £, P,(x)Qt(x) = 1. This result has the following application. The modular counting principle states that no finite set whose cardinality is not divisible by q can be partitioned into ^element classes. For each fixed cardinality N, this principle can be expressed as a propositional formula Count^fo,...) with underlying variables xe, where e ranges over <7element subsets of N. Ajtai [4] proved recently that, whenever p,q are two different primes, the propositional formulas Count $ n+I do not have polynomial size, constantdepth Frege proofs from substitution instances of Count/?, where m^O (modp). We give a new proof of this theorem based on the lower bound for Hilbert's Nullstellensatz. Furthermore our technique enables us to extend the independence results for counting principles to composite numbers p and q. This improved lower bound together with new upper bounds yield an exact characterization of when Count, can be proved efficiently from Countp, for all values of p and q.
Sparse Elimination and Applications in Kinematics
, 1994
"... This thesis proposes efficient algorithmic solutions to problems in computational algebra and computational algebraic geometry. Moreover, it considers their application to different areas where algebraic systems describe kinematic and geometric constraints. Given an arbitrary system of nonlinear mul ..."
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Cited by 47 (10 self)
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This thesis proposes efficient algorithmic solutions to problems in computational algebra and computational algebraic geometry. Moreover, it considers their application to different areas where algebraic systems describe kinematic and geometric constraints. Given an arbitrary system of nonlinear multivariate polynomial equations, its resultant serves in eliminating variables and reduces root finding to a linear eigenproblem. Our contribution is to describe the first efficient and general algorithms for computing the sparse resultant. The sparse resultant generalizes the classical homogeneous resultant and exploits the structure of the given polynomials. Its size depends only on the geometry of the input Newton polytopes. The first algorithm uses a subdivision of the Minkowski sum and produces matrix...
Matrices in Elimination Theory
, 1997
"... The last decade has witnessed the rebirth of resultant methods as a powerful computational tool for variable elimination and polynomial system solving. In particular, the advent of sparse elimination theory and toric varieties has provided ways to exploit the structure of polynomials encountered in ..."
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Cited by 44 (16 self)
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The last decade has witnessed the rebirth of resultant methods as a powerful computational tool for variable elimination and polynomial system solving. In particular, the advent of sparse elimination theory and toric varieties has provided ways to exploit the structure of polynomials encountered in a number of scientific and engineering applications. On the other hand, the Bezoutian reveals itself as an important tool in many areas connected to elimination theory and has its own merits, leading to new developments in effective algebraic geometry. This survey unifies the existing work on resultants, with emphasis on constructing matrices that generalize the classic matrices named after Sylvester, Bézout and Macaulay. The properties of the different matrix formulations are presented, including some complexity issues, with an emphasis on variable elimination theory. We compare toric resultant matrices to Macaulay's matrix and further conjecture the generalization of Macaulay's exact ratio...
What Can Be Computed in Algebraic Geometry?
 IN COMPUTATIONAL ALGEBRAIC GEOMETRY AND COMMUTATIVE ALGEBRA
, 1992
"... This paper evolved from a long series ofd;qIS3k);[ between the two authors, going back to around 1980, on the problems of making effective computations in algebraic geometry,and it took more dmorek shape in a survey talk given by the second author at a conference on Computer Algebra in 1984. The goa ..."
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Cited by 40 (0 self)
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This paper evolved from a long series ofd;qIS3k);[ between the two authors, going back to around 1980, on the problems of making effective computations in algebraic geometry,and it took more dmorek shape in a survey talk given by the second author at a conference on Computer Algebra in 1984. The goal at that time was to bring together the perspectives of theoretical computer scientists and of working algebraic geometers, while laying out what we consid)S to be the main computational problems and bound on their complexity. Only part of the talk was written d wnand since that time there has been a good dW of progress. However, the material that was written up may still serve as a useful introdok;I; to some of theidkq and estimates used in thisfield (at least the edk;;; of this volume think so), even though most of the results includk here are either published elsewhere, or exist as "folktheorems" by now. The article has four sections. The first two parts are concerned with the theory of Gröbner bases; their construction provid; the foundqIk) for most computations,and their complexity d;q;k) the complexity of most techniques in this area. The first part introdok; Gröbner bases from a geometric point of view, relating them to a number ofidS which we take up in more drekG in subsequent sections. The second part dr elops the theory of Gröbner bases more carefully, from an algebraic point of view. It could be read ind end tly,and requires less background The third part is an investigation into bound in algebraic geometry of relevance to
Efficient Algorithms and Bounds for WuRitt Characteristic Sets
 in Proc. of MEGA 90
, 1990
"... The concept of a characteristic set of an ideal was originally introduced by J.F. Ritt, in the late forties, and later, independently rediscovered by Wu WenTsun, in the late seventies. Since then WuRitt Characteristic Sets have found wide applications in Symbolic Computational Algebra, Automated T ..."
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Cited by 31 (5 self)
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The concept of a characteristic set of an ideal was originally introduced by J.F. Ritt, in the late forties, and later, independently rediscovered by Wu WenTsun, in the late seventies. Since then WuRitt Characteristic Sets have found wide applications in Symbolic Computational Algebra, Automated Theorem Proving in Elementary Geometries and Computer Vision. In this paper, we present optimal algorithms for computing a characteristic set with simpleexponential sequential and polynomial parallel time complexities. These algorithms are derived, via linear algebra, from simpleexponential degree bounds for a characteristic set. The degree bounds are obtained by using the recent effective version of Hilbert's Nullstellensatz, due to D. Brownawell and J. Koll'ar, and a version of Bezout's Inequality, due to J. Heintz. 1. Introduction In the late forties, J.F. Ritt, in his now classic book Differential Algebra [28], introduced an effective process to construct a triangular set of equations f...
A SubdivisionBased Algorithm for the Sparse Resultant
 J. ACM
, 1999
"... Multivariate resultants generalize the Sylvester resultant of two polynomials and characterize the solvability of a polynomial system. They also reduce the computation of all common roots to a problem in linear algebra. ..."
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Cited by 31 (7 self)
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Multivariate resultants generalize the Sylvester resultant of two polynomials and characterize the solvability of a polynomial system. They also reduce the computation of all common roots to a problem in linear algebra.