An Algorithmic Characterization of Polynomial Functions over $Z_{p^n}$

Abstract

In this paper we consider polynomial representability of functions defined over $Z_{p^n}$, where $p$ is a prime and $n$ is a positive integer. Our aim is to provide an algorithmic characterization that (i) answers the decision problem: to determine whether a given function over $Z_{p^n}$ is polynomially representable or not, and (ii) finds the polynomial if it is polynomially representable. The previous characterizations given by Kempner (1921) and Carlitz (1964) are existential in nature and only lead to an exhaustive search method, i.e., algorithm with complexity exponential in size of the input. Our characterization leads to an algorithm whose running time is linear in size of input. We also extend our result to the multivariate case.