Pure states, nonnegative polynomials, and sums of squares

Burgdorf, Sabine; Scheiderer, Claus; Schweighofer, Markus

Pure states, nonnegative polynomials, and sums of squares

Files

purestates.pdf(680.49 KB)

Date

2011

Authors

Burgdorf, Sabine

Scheiderer, Claus

Schweighofer, Markus

Series

Konstanzer Schriften in Mathematik; 289

Publication type

Preprint

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 Stellensätze 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.

Subject (DDC)

510 Mathematics

Keywords

pure states,extremal homomorphisms,order units,nonnegative polynomials,sums of squares,convex cones,quadratic modules,preorderings,semirings.

Cite This

ISO 690

BURGDORF, Sabine, Claus SCHEIDERER, Markus SCHWEIGHOFER, 2011. Pure states, nonnegative polynomials, and sums of squares

BibTex

@unpublished{Burgdorf2011state-15615,
  year={2011},
  title={Pure states, nonnegative polynomials, and sums of squares},
  author={Burgdorf, Sabine and Scheiderer, Claus and Schweighofer, Markus}
}

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Bibliography of Konstanz

Yes