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

TL;DR

This paper presents a new efficient algorithm to determine whether a function over the ring of integers modulo p^n can be represented by a polynomial, improving upon previous exponential-time methods and extending to multivariate functions.

Contribution

It provides a linear-time algorithmic characterization for polynomial representability over Z_{p^n}, including multivariate cases, advancing beyond prior existential characterizations.

Findings

Linear-time algorithm for polynomial representability decision

Explicit polynomial finding algorithm when representable

Extension of results to multivariate functions

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.