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

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 $\mathrm{I}\mathrm{n}\mathrm{t}(S,V)=\{f\in K[X]\mid f(S)\subseteq V\}$ 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 $\mathrm{I}\mathrm{n}\mathrm{t}(S,V)$ is Pr"ufer if and only if no element of the algebraic closure $\bar{K}$ of $K$ is a pseudo-limit of a pseudo-monotone sequence contained in $S$, with respect to some extension of $V$ to $\bar{K}$. This result expands a recent result by Loper and Werner.