  Given a function from $\mathbb{Z}_n$ to itself one can determine its
polynomial representability by using Kempner function. In this paper we present
an alternative characterization of polynomial functions over $\mathbb{Z}_n$ by
constructing a generating set for the $\mathbb{Z}_{n}$-module of polynomial
functions. This characterization results in an algorithm that is faster on
average in deciding polynomial representability. We also extend the
characterization to functions in several variables.
