Integer-valued polynomial in valued fields with an application to discrete dynamical systems

Abstract

Abstract. Integer-valued polynomials on subsets of discrete valuation domains are well studied. We undertake here a systematical study of integer-valued polynomials on subsets S of valued fields and of several connected notions: the polynomial closure of S, the Bhargava's factorial ideals of S and the v-orderings of S. A sequence of numbers is naturally associated to the subset S and a good description can be done in the case where S is regular (a generalization of the regular compact subsets of Y. Amice in local fields). Such a case arises naturally when we consider orbits under the action of an isometry.

Prüfer intersection of valuation domains of a field of rational functions

“…Unfortunately, already for a non-discrete rank one valuation domain V the precompact condition turned out to be not necessary, as Loper and Werner showed by considering subsets S of V whose elements comprise a pseudo-convergent sequence in the sense of Ostrowski (for all the definitions related to this notion see §2.1 below). It is worth recalling that the first time this notion has been used in the realm of integer-valued polynomials is in two articles of Chabert (see [8,9]). Loper and Werner made a thorough study of the rings of polynomials which are integer-valued over a pseudo-convergent sequence E = {s n } n∈N of a rank one valuation domain V , obtaining the following characterization of when Int(E, V ) is Prüfer.…”