Local polynomial functions on the integers

In 1977, H.\ Hule and W.\ N\"obauer started to investigate those unary functions on an algebra that can be interpolated at a fixed number of points by a polynomial function. In the present note, we consider the special case where the algebra is the ring of integers. For every natural number $n$, we compute a characterization of those selfmaps on the integers that can be interpolated at every subset with at most $n$ elements by a polynomial function with integral coefficients. By the results of H.\ Lausch and W.\ N\"obauer, we know that there are functions (even uncountably many of them) that can be interpolated by polynomial functions with integral coefficients at each finite subset of their domain, but fail to be polynomial functions. In this note we show that uncountably many of such ``almost polynomial'' functions exist for every countably infinite integral domain.