On the complexity of Putinar's Positivstellensatz

  • Authors:

  • Jiawang Nie;Markus Schweighofer

  • Affiliations:

  • Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA;Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany

  • Venue:
  • Journal of Complexity
  • Year:
  • 2007

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Abstract

Let S={x@?R^n|g"1(x)=0,...,g"m(x)=0} be a basic closed semialgebraic set defined by real polynomials g"i. Putinar's Positivstellensatz says that, under a certain condition stronger than compactness of S, every real polynomial f positive on S possesses a representation f=@?"i"="0^m@s"ig"i where g"0@?1 and each @s"i is a sum of squares of polynomials. Such a representation is a certificate for the nonnegativity of f on S. We give a bound on the degrees of the terms @s"ig"i in this representation which depends on the description of S, the degree of f and a measure of how close f is to having a zero on S. As a consequence, we get information about the convergence rate of Lasserre's procedure for optimization of a polynomial subject to polynomial constraints.