A complete classification of $\mathbb{Z}_{p}$-sequences corresponding to a polynomial

Abstract

Let $p$ be a prime number and set ${\mathbb{Z}}_p=\mathbb{Z}∕p\mathbb{Z}$. A ${\mathbb{Z}}_p$-sequence is a function $S:\mathbb{Z}\to {\mathbb{Z}}_p$. Let $\mathcal{ℛ}$ be the set $\left\{P\in ℝ\left[X\right]\mid P\left(\mathbb{Z}\right)\subseteq \mathbb{Z}\right\}$. We prove that the set of sequences of the form ${\left(P\left(n\right)\left(\text{ mod }p\right)\right)}_{n\in \mathbb{Z}}$, where $P\in \mathcal{ℛ}$, is precisely the set of periodic ${\mathbb{Z}}_p$-sequences with period equal to a $p$-power. Given a ${\mathbb{Z}}_p$-sequence, we will also determine all $P\in \mathcal{ℛ}$ that correspond to the sequence according to the manner above.