On the inverse problem of the $k$-th Davenport constants for groups of rank $2$
Qinghai Zhong
TL;DR
This work addresses the inverse problem for the $k$-th Davenport constant $\mathsf D_k(G)$ in rank-2 finite abelian groups $G \cong C_{n_1} \oplus C_{n_2}$ with $1< n_1 \mid n_2$ and $n_1<n_2$, building on the known formula $\mathsf D_k(G)=n_1+k n_2-1$. It provides a complete characterization of extremal zero-sum sequences $U$ of length $|U|=\mathsf D_k(G)$ that cannot be partitioned into $k+1$ nontrivial zero-sum subsequences, listing four explicit forms (I–IV) that such sequences must take. The proof relies on a reduction to quotients $G/H$ with cyclic $H$ and a detailed block-decomposition analysis, together with projection to a quotient isomorphic to $\varphi(G)\cong C_n\oplus C_n$ and subsequent lifting back to $G$. The results advance inverse zero-sum theory for rank-2 groups and provide concrete templates for extremal sequences with potential applications in invariant theory and related combinatorial number theory problems.
Abstract
For a finite abelian group $G$ and a positive integer $k$, let $\mathsf{D}_k(G)$ denote the smallest integer $\ell$ such that each sequence over $G$ of length at least $\ell$ has $k$ disjoint nontrivial zero-sum subsequences. It is known that $\mathsf D_k(G)=n_1+kn_2-1$ if $G\cong C_{n_1}\oplus C_{n_2}$ is a rank $2$ group, where $1<n_1\t n_2$. We investigate the associated inverse problem for rank $2$ groups, that is, characterizing the structure of zero-sum sequences of length $\mathsf D_k(G)$ that can not be partitioned into $k+1$ nontrivial zero-sum subsequences.