A Multiplicative Property for Zero-Sums II

Abstract

For n ≥ 1, let Cn denote a cyclic group of order n. Let G∼= Cn ⊕Cmn with n ≥ 2 and m ≥ 1, and let k ∈ [0, n − 1]. It is known that any sequence of mn + n − 1 + k terms from G must contain a nontrivial zero-sum of length at most mn + n − 1 − k. The associated inverse question is to characterize those sequences with maximal length mn + n − 2 + k that fail to contain a nontrivial zero-sum subsequence of length at most mn + n − 1 − k. For k ≤ 1, this is the inverse question for the Davenport Constant. For k = n − 1, this is the inverse question for the η(G) invariant concerning short zero-sum subsequences. For Cn ⊕ Cn and k ∈ [2, n − 2], with n ≥ 5 prime, it was conjectured in a paper of Grynkiewicz, Wang and Zhao that they must have the form S = e[n−1] 1 · e[n−1] 2 · (e1 + e2)[k] for some basis (e1, e2), with the conjecture established in many cases and later extended to composite moduli n. In this paper, we focus on the case m ≥ 2. Assuming the conjectured structure holds for k ∈ [2, n − 2] in Cn ⊕ Cn, we characterize the structure of all sequences of maximal length mn + n − 2 + k in Cn ⊕ Cmn that fail to contain a nontrivial zero-sum of length at most mn + n − 1 − k, showing they must either have the form S = e[n−1] 1 · e[sn−1] 2 · (e1 + e2)[(m−s)n+k] for some s ∈ [1, m] and basis (e1, e2) with ord(e2) = mn, or else have the form S = g[n−1] 1 · g[n−1] 2 · (g1 + g2)[(m−1)n+k] for some generating set {g1, g2} with ord(g1 + g2) = mn. In view of prior work, this reduces the structural characterization for a general rank two abelian group to the case Cp ⊕ Cp with p prime. Additionally, we give a new proof of the precise structure in the case k = n−1 for m = 1. Combined with known results, our results unconditionally establish the structure of extremal sequences in G∼= Cn ⊕ Cmn in many cases, including when n is only divisible by primes at most 7, when n ≥ 2 is a prime power and k ≤2n+1 3, or when n is composite and k = n − d − 1 or n − 2d + 1 for a proper, nontrivial divisor d | n.

Publication Title

Electronic Journal of Combinatorics

Link to Full Text

Share

COinS