Submission date: 29 June 2018

Abstract:

Inspired by logical diophantine problems, we regard discrete orderings as “arithmetical” points living in a real spectrum space and prove a theorem on the distribution of the “arithmetical” points in the affine space in terms of its geometry. To be precise, let M be a discretely ordered ring and R be a real closed field containing M, we prove that any ball B(a,r) in Sper(R[X_1,...,X_n]) with center a and radius r (defined via Robson's metric) contains a discrete ordering of M[X_1,...,X_n] whenever r is non-infinitesimal and a is away from all hyperplanes over M.

Mathematics Subject Classification: 03H15, 14P99

Keywords and phrases: Real spectrum, Discrete orderings

Full text arXiv 1807.00501: pdf, ps.