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.Comment: to appear in J. Algebra. All comments are welcome. Keywords: Pr\"ufer
  domain, pseudo-convergent sequence, pseudo-limit, residually transcendental
  extension, integer-valued polynomia

Peruginelli, Giulio

English

arXiv

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 $\Int(S,V)=\{f\in K[X] \mid f(S)\subseteq V\}$ 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 $\Int(S,V)$ 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

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

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