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

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 $a_1,\ldots,a_h$ and negative terms $b_1,\ldots,b_k$. We prove that $h\leq \lfloor σ^+/k\rfloor$ and $k\leq \lfloor σ^+/h\rfloor$, where $σ^+=\sum_{i=1}^h a_i=-\sum_{j=1}^k b_j$. 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 $\{i\in {\mathbb Z}:\; -n\leq i\leq n\}$ for any positive integer $n$.