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

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 $a_1,\dots ,a_h$ and negative terms $b_1,\dots ,b_k$. We prove that $h\le \lfloor {\sigma }^{+}/k\rfloor$ and $k\le \lfloor {\sigma }^{+}/h\rfloor$, where ${\sigma }^{+}={\sum }_{i=1}^h{a}_i=-{\sum }_{j=1}^k{b}_j$. 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 $\{i\in \mathbb{Z}:-n\le i\le n\}$ for any positive integer $n$.

10 pages, 1 fugure; to appear in Acta Arithmetica