A new proof for the existence of degree bounds for Putinar's Positivstellensatz

TL;DR

This paper presents a new, simple, and non-constructive proof for the existence of degree bounds in Putinar's Positivstellensatz, a key result in real algebraic geometry with applications in polynomial optimization.

Contribution

It introduces an elementary and concise proof for degree bounds, improving understanding and potentially simplifying applications in polynomial optimization.

Findings

Provides a new elementary proof for degree bounds

Simplifies the theoretical understanding of Putinar's Positivstellensatz

Potentially impacts polynomial optimization methods

Abstract

Putinar's Positivstellensatz is a central theorem in real algebraic geometry. It states the following: If you have a set $S= \{ x \in R^n \ | \ g_1 (x) \geq 0, ... , g_m(x) \geq 0\}$ described by some real polynomials $g_i$, then every real polynomial $f$ that is positive on $S$ can be written as a sum of squares weighted by the $g_i$ and $1$. Consider such an identity $f= \sum_{i=1}^{m} g_i s_i + s_0$. For the applications in polynomial optimization, especially semidefinite programming, the following is important: There exists a bound $N$ for the degrees of the $s_i$ which depends only on the $g_i$, $n$, the degree of $f$, an upper bound for $||f||$ and a lower bound for $\min f(S)$. Two proofs from Prestel and He{\ss} resp. Schweighofer and Nie ([Pr], [He] resp. [Sw], [NS]) for the existence of these degree bounds are known (also for the matrix version of Putinar's Positivstellensatz by Helton and Nie [HN]). Prestel uses valuation and model theory for his approach while Schweighofer gives a constructive solution by using a theorem of P\'{o}lya. In this paper we will give a new elementary, short but non-constructive proof.