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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 222 (18 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
"... ..."
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.
Hilbert's Nullstellensatz is in the Polynomial Hierarchy
 Journal of Complexity
, 1996
"... We show that if the Generalized Riemann Hypothesis is true, the problem of deciding whether a system of polynomial equations in several complex variables has a solution is in the second level of the polynomial hierarchy. In fact, this problem is in AM, the "ArthurMerlin" class (recall that NP ` AM ..."
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Cited by 40 (9 self)
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We show that if the Generalized Riemann Hypothesis is true, the problem of deciding whether a system of polynomial equations in several complex variables has a solution is in the second level of the polynomial hierarchy. In fact, this problem is in AM, the "ArthurMerlin" class (recall that NP ` AM ` RP NP ` \Pi 2 ). The best previous bound was PSPACE. An earlier version of this paper was distributed as NeuroCOLT Technical Report 9644. The present paper includes in particular a new lower bound for unsatisfiable systems, and remarks on the ArthurMerlin class. 1 A part of this work was done when the author was visiting DIMACS at Rutgers University. 1 Introduction In its weak form, Hilbert's Nullstellensatz states that a system f 1 (x) = 0; : : : ; f s (x) = 0 (1) of polynomial equations in n unknowns has no solution over C if and only if there are polynomials g 1 ; : : : ; g s 2 C [X 1 ; : : : ; X n ] such that P s i=1 f i g i = 1. For this reason, the problem of deciding whethe...
Polar Varieties and Efficient Real Equation Solving: The Hypersurface Case
 PROCEEDINGS OF THE 3RD CONFERENCE APPROXIMATION AND OPTIMIZATION IN THE CARIBBEAN, IN: APORTACIONES MATEMÁTICAS, MEXICAN SOCIETY OF MATHEMATICS
, 1998
"... The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo [10] can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bou ..."
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Cited by 26 (6 self)
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The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo [10] can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bounded smooth hypersurface. The algorithm in [10] yields a method for symbolically solving a zerodimensional polynomial equation system in the affine (and toric) case. Its main feature is the use of adapted data structure: Arithmetical networks and straightline programs. The algorithm solves any affine zerodimensional equation system in nonuniform sequential time that is polynomial in the length of the input description and an adequately defined affine degree of the equation system. Replacing the affine degree of the equation system by a suitably defined real degree of certain polar varieties associated to the input equation, which describes the hypersurface under consideration, and using straightline program codification of the input and intermediate results, we obtain a method for the problem introduced above that is polynomial in the input length and the real degree.
A survey of the Slemma
 SIAM Review
"... Abstract. In this survey we review the many faces of the Slemma, a result about the correctness of the Sprocedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as ..."
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Cited by 26 (0 self)
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Abstract. In this survey we review the many faces of the Slemma, a result about the correctness of the Sprocedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as well. These were all active research areas, but as there was little interaction between researchers in these different areas, their results remained mainly isolated. Here we give a unified analysis of the theory by providing three different proofs for the Slemma and revealing hidden connections with various areas of mathematics. We prove some new duality results and present applications from control theory, error estimation, and computational geometry. Key words. Slemma, Sprocedure, control theory, nonconvex theorem of alternatives, numerical range, relaxation theory, semidefinite optimization, generalized convexities
Sharp estimates for the arithmetic Nullstellensatz
 Duke Math. J
"... We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over the integer ring Z. The result improves previous work of P. Philippon, C. Berenstein and A. Yger, and T. Krick and L. M. Pardo. We also present degree and height estimates of intrinsic type, which ..."
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Cited by 25 (2 self)
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We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over the integer ring Z. The result improves previous work of P. Philippon, C. Berenstein and A. Yger, and T. Krick and L. M. Pardo. We also present degree and height estimates of intrinsic type, which depend mainly on the degree and the height of the input polynomial system. As an application we derive an effective arithmetic Nullstellensatz for sparse polynomial systems. The proof of these results relies heavily on the notion of local height of an affine
Sharp estimates for the arithmetic
 Nullstellensatz, Duke Math. J. 109
, 2001
"... Nullstellensatz ..."
Some Complexity Results for Polynomial Ideals
, 1997
"... In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + 1)tuple P = ( f, g1, g2,.. ..."
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Cited by 19 (0 self)
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In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + 1)tuple P = ( f, g1, g2,..., gw) where f and the gi are multivariate polynomials, and the problem is to determine whether f is in the ideal generated by the gi. For polynomials over the integers or rationals, this problem is known to be exponential space complete. We discuss further complexity results for problems related to polynomial ideals, like the word and subword problems for commutative semigroups, a quantitative version of Hilbert’s Nullstellensatz in a complexity theoretic version, and problems concerning the computation of reduced polynomials and Gröbner bases.