A complete classification of ℤp-sequences corresponding to a polynomial

Huang, Leonard

doi:10.2140/involve.2009.2.411

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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

DOI: 10.2140/involve.2009.2.411

Abstract

Let $p$ be a prime number and set $ℤ_{p} = ℤ ∕ p ℤ$ . A $ℤ_{p}$ -sequence is a function $S : ℤ \to ℤ_{p}$ . Let $ℛ$ be the set ${P \in ℝ [X] ∣ P (ℤ) \subseteq ℤ}$ . We prove that the set of sequences of the form ${(P (n) (mod p))}_{n \in ℤ}$ , where $P \in ℛ$ , 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 $P \in ℛ$ that correspond to the sequence according to the manner above.

Keywords

$\mathbb{Z}_p$-sequences, polynomials, free abelian group

Mathematical Subject Classification 2000

Primary: 11B83

Milestones

Received: 25 October 2008

Revised: 15 September 2009

Accepted: 15 September 2009

Published: 28 October 2009

Communicated by Andrew Granville

Authors

Leonard Huang
School of Physical and Mathematical Sciences Nanyang Technological University SPMS-04-01 21 Nanyang Link 637371 Singapore

Vol. 2, No. 4, 2009