On the Davenport constant and group algebras

Abstract

For a finite abelian group $G$ and a splitting field $K$ of $G$, let $d(G,K)$ denote the largest integer $l\in \mathbb{N}$ for which there is a sequence $S=g_1\cdot ...\cdot g_l$ over $G$ such that $(X^{g_1}-a_1)\cdot ...\cdot (X^{g_l}-a_l)\ne 0\in K[G]$ for all $a_1,...,a_l\in K^{\times }$. If $D(G)$ denotes the Davenport constant of $G$, then there is the straightforward inequality $D(G)-1\le d(G,K)$. Equality holds for a variety of groups, and a standing conjecture of W. Gao et.al. states that equality holds for all groups. We offer further groups for which equality holds, but we also give the first examples of groups $G$ for which $D(G)-1<d(G,K)$ holds. Thus we disprove the conjecture.

12 pages; fixed typos and clearer proof of Lemma 3.9

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