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[X1,…,Xn] in Specr(R[X1,…,Xn]) in geometric terms. To be more precise, we prove that any ball B(α,r) in Specr(R[X1,…,Xn]) with center α and radius r (defined via Robson's metric) contains a discrete ordering of M[X1,…,Xn] whenever r is positive non-infinitesimal and α is at infinite distance from all hyperplanes over M.
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