Discrete orderings in the real spectrum

TL;DR

This paper explores the structure of discrete orderings within the real spectrum of a ring, providing geometric interpretations and distribution results for these orderings in polynomial rings over discretely ordered rings.

Contribution

It introduces the concept of discrete prime cones and establishes a geometric distribution theorem for discrete orderings in polynomial rings over discretely ordered rings.

Findings

Discrete prime cones are defined with an algebro-geometric interpretation.

Any non-infinitesimal ball in the spectrum contains a discrete ordering.

Distribution of discrete orderings relates to geometric properties of the spectrum.

Abstract

We study discrete orderings in the real spectrum of a commutative ring by defining discrete prime cones and give an algebro-geometric meaning to some kind of diophantine problems over discretely ordered rings. Also for a discretely ordered ring $M$ and a real closed field $R$ containing $M$ we prove a theorem on the distribution of the discrete orderings of $M[X_1,\dots,X_n]$ in $\Spec(R[X_1,\dots,X_n])$ in geometric terms. To be more precise, we prove that any ball $\mathbb{B}(\alpha,r)$ in $\Spec(R[X_1,\dots,X_n]$) with center $\alpha$ and radius $r$ (defined via Robson's metric) contains a discrete ordering of $M[X_1,\dots,X_n]$ whenever $r$ is non-infinitesimal and $\alpha$ is away from all hyperplanes over $M$ passing through the origin.