Finite differences in finite characteristic

Abstract

Let D be a Dedekind domain with characteristic p > 0 . In this paper, we are interested in the D-algebras Int [ k ] ( D ) of integer-valued polynomials with all their k-first finite differences. Mainly, we describe the characteristic ideals J n [ k ] ( D ) of Int [ k ] ( D ) . When D = V is the ring of a discrete valuation domain, we correct an old formula given by Barsky and we construct bases of the V-module Int [ k ] ( V ) . When D = F q [ T ] , we give a more explicit formula for the J n [ k ] ( F q [ T ] ) 's and describe a new basis for Int [ k ] ( F q [ T ] ) that comes from a regular basis of Int ( F q [ T ] ) introduced by M. Car.