Schm\"udgen's theorem and results of positivity

TL;DR

This paper provides new proofs of well-known Positivstellensätze using the Stone-Weierstrass Theorem, leading to homogeneous and projective versions of Schmüdgen's and Putinar's theorems, simplifying existing approaches.

Contribution

It introduces a novel proof method that bypasses the Kadison-Dubois Theorem and extends Positivstellensätze to homogeneous and projective cases.

Findings

Homogeneous version of Schmüdgen's theorem established

Projective version of Putinar's theorem derived

Simplified proofs of classical Positivstellensätze

Abstract

This is a translation of a final paper. It contains proofs of some well-known Positivstellens\"atze. The approach in this paper differs in two respects from presentations in literature. We use the Stone-Weierstrass Theorem directly to prove an inductive property, which makes the Kadison-Dubois Theorem superfluous in our paper, and we also consider homogeneous and projective Positivstellens\"atze. This leads us to a homogeneous version of Schm\"udgen's theorem and a projective version of Putinar's theorem.