On central orderings

TL;DR

This paper introduces the concept of central orderings for commutative rings, extending the idea of central points in real algebraic geometry, and explores their properties to formulate Positivestellensätze akin to Hilbert's 17th problem.

Contribution

It generalizes the notion of central points to arbitrary commutative rings and develops a framework for central and precentral loci in the real spectrum.

Findings

Defined central orderings for commutative rings.

Established properties of central and precentral loci.

Formulated central Positivestellensätze similar to Hilbert's 17th problem.

Abstract

We define the notion of central orderings for a general commutative ring $A$ which generalizes the notion of central points of irreducible real algebraic varieties. We study a central and a precentral loci which both live in the real spectrum of the ring $A$ and allow to state central Positivestellens\"atze in the spirit of Hilbert 17th problem.