On the existence of distinct lengths zero-sum subsequences

Details On the existence of distinct lengths zero-sum subsequences On the existence of distinct lengths zero-sum subsequences

2009-03-20

arxiv.org/abs/0903.3458v3

Mathematics

Number Theory

10 pages, to appear in Rocky Mountain Journal of Mathematics

Scientific paper

In this paper, we obtain a characterization of short normal sequences over a finite Abelian p-group, thus answering positively a conjecture of W. Gao for a variety of such groups. Our main result is deduced from a theorem of N. Alon, S. Friedland and G. Kalai, originally proved so as to study the existence of regular subgraphs in almost regular graphs. In the special case of elementary p-groups, Gao's conjecture is solved using N. Alon's Combinatorial Nullstellensatz. To end with, we show that, assuming every integer satisfies Property B, this conjecture holds in the case of finite Abelian groups of rank two.

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