#### Vol. 2, No. 4, 2009

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

### Leonard Huang

Vol. 2 (2009), No. 4, 411–418
##### Abstract

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

##### Keywords
$\mathbb{Z}_p$-sequences, polynomials, free abelian group
Primary: 11B83