A Note on Minimal zero-sum sequences over \mathbb Z

2014/01/03 by Papa A. Sissokho, Sissokho, Papa A.
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary: 11B75 #math.CO #math.NT #msc:11B75

paper · pdf · doi:10.48550/arxiv.1401.0715

10 pages, 1 fugure; to appear in Acta Arithmetica

arxiv created 2014/07/26 · arxiv updated 2014/07/29

Abstract

A zero-sum sequence over \mathbb Z is a sequence with terms in \mathbb Z that sum to 0. It is called minimal if it does not contain a proper zero-sum subsequence. Consider a minimal zero-sum sequence over \mathbb Z with positive terms a₁,…,a_h and negative terms b₁,…,b_k. We prove that h≤ \lfloor σ⁺/k\rfloor and k≤ \lfloor σ⁺/h\rfloor, where σ⁺=∑_i=1^h a_i=-∑_j=1^k b_j. These bounds are tight and improve upon previous results. We also show a natural partial order structure on the collection of all minimal zero-sum sequences over the set \i∈ \mathbb Z: -n≤ i≤ n\ for any positive integer n.

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