Minimal zero-sum sequences in \(C_{n}\oplus C_{n}\)
By a result of \textit{J. E. Olson} [``A combinatorial problem on finite abelian groups. II'', J. Number Theory 1, 195--199 (1969; Zbl 0167.28004)], any sequence of elements of \(G=C_n\oplus C_n\) with length at least \(2(n-1)+1\) contains a nonempty zero-sum subsequence, where \(C_n\) is the additive cyclic group of order \(n\). Let \(\{g_i\}_{i=1}^{2n-1}\) be a minimal zero-sum sequence with \(g_i\in G\). \textit{W. D. Gao} and \textit{A. Geroldinger} [``On zero-sum sequences in \(\mathbb Z/n\mathbb Z\oplus\mathbb Z/n\mathbb Z\),'' Integers 3, Paper A08, 45 p., electronic only (2003; Zbl 1078.11010)] conjectured that the exact value of \(M=\max_{g\in G}| \{1\leq i\leq 2n-1:\, g_i=g\}| \) is \(n-1\); when \(n\) is an odd prime \(p\), they showed that \(3\leq| \{g_1,\ldots,g_{2p-1}\}| \leq p+1\), which is a consequence of the conjectured equality \(M=p-1\). In the paper under review, the authors obtain the further inequality \(| \{g_1,\ldots,g_{2p-1}\}| \leq p\).
scientific article; zbMATH DE number 5137580
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