On the complexity of Putinar's Positivstellensatz

TL;DR

This paper establishes an upper bound on the degree complexity of Putinar's Positivstellensatz, providing insights into the convergence rate of Lasserre's polynomial optimization method.

Contribution

It offers a new degree complexity bound for Putinar's Positivstellensatz, highlighting differences with Schm"udgen's approach and implications for polynomial optimization convergence.

Findings

Degree complexity bound for Putinar's Positivstellensatz

Comparison with Schm"udgen's Positivstellensatz bounds

Implications for Lasserre's polynomial optimization convergence

Abstract

We prove an upper bound on the degree complexity of Putinar's Positivstellensatz. This bound is much worse than the one obtained previously for Schm\"udgen's Positivstellensatz but it depends on the same parameters. As a consequence, we get information about the convergence rate of Lasserre's procedure for optimization of a polynomial subject to polynomial constraints.