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A Faster Algorithm For Testing Polynomial Representability Of Functions Over Finite Integer Rings

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.