Discrete orderings in the real 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.