Constructive proofs of some positivstellens\"atze for compact semialgebraic subsets of $\mathbb{R}^d$

TL;DR

This paper provides constructive and simplified proofs of certain positivstellensätze, which are algebraic certificates of positivity for polynomials on compact semialgebraic sets, enhancing understanding and accessibility.

Contribution

It introduces elementary, constructive proofs for known positivstellensätze, extending and simplifying previous arguments by Berr, Wörmann, and Schweighofer.

Findings

Constructive proofs for positivstellensätze on compact semialgebraic sets.

Simplification and extension of existing proof techniques.

Enhanced accessibility of algebraic positivity certificates.

Abstract

In a broad sense, positivstellens\"atze are results about representations of polynomials which are strictly positive on a given set. We give constructive and, to a large extent, elementary proofs of some known positivstellens\"atze for compact semialgebraic subsets of $\mathbb{R}^d$. The presented proofs extend and simplify arguments of Berr, W\"ormann (2001) and Schweighofer (2002, 2005).