The maximal-length minimal zero-sum sequence conjecture

Let GG be a finite abelian group, and call a sequence over GG a minimal zero-sum sequence if its sum is zero and no nonempty proper subsum is zero. Write D(G)\mathsf D(G) for the Davenport constant, the maximal length of a minimal zero-sum sequence over GG. The maximal-length minimal zero-sum sequence conjecture. If GG is neither cyclic nor an elementary 22-group, then for every minimal zero-sum sequence U=g1gU=g_1\ldots g_\ell with U==D(G)|U|=\ell=\mathsf D(G), there exist kNk\in\mathbb N and minimal zero-sum sequences U1,,Uk,V1,,Vk+1U_1,\ldots,U_k,V_1,\ldots,V_{k+1}, whose terms belong to {g1,,g,g1,,g}\{g_1,\ldots,g_\ell,-g_1,\ldots,-g_\ell\}, such that

U1Uk=V1Vk+1.U_1\ldots U_k=V_1\ldots V_{k+1}.

A positive answer gives a structural description of sets of lengths with maximal elasticity in transfer Krull monoids over finite abelian groups. The supplied text does not state whether this conjecture has been resolved.

Sources & referencesView supporting material

Primary source

Aqsa Bashir, Alfred Geroldinger and Qinghai Zhong, “On a zero-sum problem arising from factorization theory”, arXiv:2007.10094 (2020).

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