Pure states, nonnegative polynomials and sums of squares

TL;DR

This paper introduces pure states into commutative algebra to simplify proofs of positivity certificates for polynomials on semialgebraic sets, extending results to polynomials with arbitrary zeros.

Contribution

It provides a new, more conceptual approach to archimedean Positivstellensätze and establishes novel results for polynomials with non-discrete zeros.

Findings

Simplified proofs of key Positivstellensätze using pure states

Extended results to polynomials with arbitrary zeros

New theorems allowing non-discrete zeros on semialgebraic sets

Abstract

In recent years, much work has been devoted to a systematic study of polynomial identities certifying strict or non-strict positivity of a polynomial on a basic closed semialgebraic set. The interest in such identities originates not least from their importance in polynomial optimization. The majority of the important results requires the archimedean condition, which implies that the semialgebraic set has to be compact. This paper introduces the technique of pure states into commutative algebra. We show that this technique allows an approach to most of the recent archimedean Stellensaetze that is considerably easier and more conceptual than the previous proofs. In particular, we reprove and strengthen some of the most important results from the last years. In addition, we establish several such results which are entirely new. They are the first that allow the polynomial to have arbitrary, not necessarily discrete, zeros on the semialgebraic set.