A note on minimal zero-sum sequences over $\mathbb Z$

Volume 166 / 2014

Papa A. Sissokho Acta Arithmetica 166 (2014), 279-288 MSC: Primary 11B75; Secondary 11B30, 11P70. DOI: 10.4064/aa166-3-4


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 $n$.


  • Papa A. SissokhoMathematics Department
    Illinois State University
    Normal, IL 61790-4520, U.S.A.

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