Title: Valuation domains of the field of rational functions associated to pseudo-convergent sequences

**Speaker:** Giulio Peruginelli - University of Padova

**Abstract**: Let V be a valuation domain of rank one with quotient field K. Recently, Loper and Werner gave a natural construction of a valuation domain V_E of the field of rational function K(X) lying over V associated to a pseudo-convergent sequence E=\{s_n\}_{n\in\mathbb{N}} of K, when E is either of transcendental type or of algebraic type and zero-breadth ideal. This

construction has been employed by Loper and Werner in the realm of integer-valued polynomials. We show that their construction works for any valuation domain V and for any pseudo-convergent sequence of K. We also show that, when V has rank one, the valuation domain V_E is strictly related to another extension of V to K(X) as given by Ostrowski in 1935. We give some results about the Zariski-Riemann spaces of such valuation domains. This is a joint work with Dario Spirito.

If time allows, I will also introduce the more general definition of pseudo-monotone sequence as given by Chabert, which has been an essential tool to establish when the ring of integer-valued polynomials over a subset S of V, that is \textnormal{Int}(S,V)=\{f\in K[X] \mid f(S)\subseteq V\}, is a Pr\"ufer domain.