# The structure of maximal zero-sum free sequences

@article{Bhowmik2008TheSO, title={The structure of maximal zero-sum free sequences}, author={Gautami Bhowmik and Immanuel Halupczok and Jan-Christoph Schlage-Puchta}, journal={Acta Arithmetica}, year={2008}, volume={143}, pages={21-50} }

Let n be an integer, and consider finite sequences of elements of the group Z/nZ x Z/nZ. Such a sequence is called zero-sum free, if no subsequence has sum zero. It is known that the maximal length of such a zero-sum free sequence is 2n-2, and Gao and Geroldinger conjectured that every zero-sum free sequence of this length contains an element with multiplicity at least n-2. By recent results of Gao, Geroldinger and Grynkiewicz, it essentially suffices to verify the conjecture for n prime. Now…

## 13 Citations

### Long Zero-Sum Free Sequences over Cyclic Groups

- Mathematics, Biology
- 2013

The goal of this chapter is characterize the structure of those zero-sum free sequences close to the extremal possible length, and will be able to characterize this structure for sequences quite a ways away from the maximal value.

### Inverse zero-sum problems III

- Mathematics
- 2008

Let G be a nite abelian group. The Davenport constant D(G) is the smallest integer ` 2 N such that every sequence S over G of length jSj ` has a nontrivial zero-sum subsequence. Let G = Cn Cn with n…

### Inverse Zero-Sum Problems III: Addendum

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- 2021

The Davenport constant for a finite abelian group G is the minimal length l such that any sequence of l terms from G must contain a nontrivial zero-sum sequence. For the group G = (Z/nZ), its value…

### Structure of a sequence with prescribed zero-sum subsequences: Rank Two $p$-groups

- Mathematics
- 2022

. Let G = ( Z /n Z ) ⊕ ( Z /n Z ). Let s ≤ k ( G ) be the smallest integer ℓ such that every sequence of ℓ terms from G , with repetition allowed, has a nonempty zero-sum subsequence with length at…

### Inverse zero-sum problems II

- Mathematics
- 2008

Let $G$ be an additive finite abelian group. A sequence over $G$ is called a minimal zero-sum sequence if the sum of its terms is zero and no proper subsequence has this property. Davenport's…

### The Inverse Problem Associated to the Davenport Constant for C2+C2+C2n, and Applications to the Arithmetical Characterization of Class Groups

- MathematicsElectron. J. Comb.
- 2011

A characterization, via the system of sets of lengths, of the class group of rings of algebraic integers is obtained for certain types of groups, and the Davenport constants of groups of the form C 4 ⊕ C4n and C 6 ⊵ C6n are determined.

### Arithmetic-progression-weighted subsequence sums

- Mathematics
- 2011

AbstractLet G be an abelian group, let s be a sequence of terms s1, s2, …, sn ∈ G not all contained in a coset of a proper subgroup of G, and let W be a sequence of n consecutive integers. Let
$$W…

### Products of two atoms in Krull monoids and arithmetical characterizations of class groups

- MathematicsEur. J. Comb.
- 2013

### A Multiplicative Property for Zero-Sums II

- MathematicsElectron. J. Comb.
- 2022

For $n\geq 1$, let $C_n$ denote a cyclic group of order $n$. Let $G=C_n\oplus C_{mn}$ with $n\geq 2$ and $m\geq 1$, and let $k\in [0,n-1]$. It is known that any sequence of $mn+n-1+k$ terms from $G$…

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Let G be a nite abelian group. The Davenport constant D(G) is the smallest integer ` 2 N such that every sequence S over G of length jSj ` has a nontrivial zero-sum subsequence. Let G = Cn Cn with n…

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