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 $h\leq \lfloor \sigma^+/k\rfloor$ and $k\leq \lfloor \sigma^+/h\rfloor$, 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 $\{ i\in { \mathbb Z}:\; -n\leq i\leq n\}$ for any positive integer $n$.
Categories: math.CO math.NT