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

Papa A. Sissokho¹

¹ Mathematics Department Illinois State University Normal, IL 61790-4520, U.S.A.

Acta Arithmetica, Tome 166 (2014) no. 3, pp. 279-288

Résumé

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 $n$.

Détail

DOI : 10.4064/aa166-3-4

Keywords: zero sum sequence mathbb sequence terms mathbb sum nbsp called minimal does contain proper zero sum subsequence consider minimal zero sum sequence mathbb positive terms ldots negative terms ldots prove leq lfloor sigma rfloor leq lfloor sigma rfloor where sigma sum sum these bounds tight improve previous results natural partial order structure collection minimal zero sum sequences set mathbb n leq leq positive integer