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

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.