Pr\"ufer intersection of valuation domains of a field of rational functions

TL;DR

This paper characterizes when the ring of integer-valued polynomials over a subset of a valuation domain is a Pr"ufer domain, using generalized notions of pseudo-monotone sequences and their pseudo-limits.

Contribution

It introduces a characterization of Pr"ufer integer-valued polynomial rings via pseudo-monotone sequences, extending previous results by Loper and Werner.

Findings

${ m Int}(S,V)$ is Pr"ufer iff no algebraic closure element is a pseudo-limit of a pseudo-monotone sequence in $S$

The characterization generalizes classical notions of pseudo-convergence and pseudo-limits

Provides a criterion linking valuation theory and polynomial rings over subsets

Abstract

Let $V$ be a rank one valuation domain with quotient field $K$. We characterize the subsets $S$ of $V$ for which the ring of integer-valued polynomials ${\rm Int}(S,V)=\{f\in K[X] \mid f(S)\subseteq V\}$ is a Pr\"ufer domain. The characterization is obtained by means of the notion of pseudo-monotone sequence and pseudo-limit in the sense of Chabert, which generalize the classical notions of pseudo-convergent sequence and pseudo-limit by Ostrowski and Kaplansky, respectively. We show that ${\rm Int}(S,V)$ is Pr\"ufer if and only if no element of the algebraic closure $\overline{K}$ of $K$ is a pseudo-limit of a pseudo-monotone sequence contained in $S$, with respect to some extension of $V$ to $\overline{K}$. This result expands a recent result by Loper and Werner.