On the existence of zero-sum subsequences of distinct lengths (Q422051)

Let \((G,+)\) denote a finite abelian group. The Davenport constant \(\mathsf{D}(G)\) is the smallest integer such that each sequence of elements of \(G\) of that length has a nonempty subsequence whose terms sum to \(0\), the neutral element of the group.NEWLINENEWLINEThe present paper studies a related question, more specifically a conjecture of Gao.NEWLINENEWLINEA sequence \(S\) over \(G\) of length at least \(\mathsf{D}(G)\) is called normal if every subsequence of \(S\) that has sum equal to \(0\) has length at most \(|S| - (\mathsf{D}(G) - 1) \). It is easy to construct normal sequences of arbitrary length; just take a sequence \(T\) of length \(\mathsf{D}(G) - 1\) without any subsequence with sum \(0\), called a zero-sum free sequence, and augment this sequence by any number of \(0\)'s --- the existence of \(T\) is guaranteed by the definition of \(\mathsf{D}(G) - 1\).NEWLINENEWLINEThe question raised by Gao is whether this simple construction already yields all normal sequences (under some additional assumption). More precisely, he conjectured that for \(G\) isomorphic to \(C_{n_1} \oplus \dots \oplus C_{n_r}\), with \(n_1 \mid \dots \mid n_r\) and where \(C_{n_i}\) denotes a cyclic group of order \(n_i\), each normal sequence of length \(\mathsf{D}(G) + j - 1\) with \(1 \leq j \leq n_1-1\) can be obtained via the above construction.NEWLINENEWLINEThis conjecture had been confirmed in special cases. In the present paper it is confirmed for additional groups, namely \(p\)-groups where \(n_1\) is a prime and groups of rank two.NEWLINENEWLINEFor the former an interesting result on the existence of subsequences with sum \(0\) whose lengths fulfill certain congruence conditions is obtained, using the Combinatorial Nullstellensatz. For the latter, the characterization of zero-sum free sequences of maximal length is used; note tha when the paper was written this characterization was only known conditionally but, as commented in addendum in the paper, is meanwhile known unconditionally.NEWLINENEWLINEIn addition two conjectures on related problems are formulated.