On the Davenport constant and group algebras
Volume 121 / 2010
Abstract
For a finite abelian group $G$ and a splitting field $K$ of $G$, let $\mathsf d (G, K)$ denote the largest integer $l \in \mathbb N$ for which there is a sequence $S = g_1 \cdot \ldots \cdot g_l$ over $G$ such that $(X^{g_1} - a_1) \cdot \ldots \cdot (X^{g_l} - a_l) \ne 0 \in K[G]$ for all $a_1, \ldots, a_l \in K^{\times}$. If $\mathsf D (G)$ denotes the Davenport constant of $G$, then there is the straightforward inequality $\mathsf D (G)-1 \le \mathsf d (G, K)$. Equality holds for a variety of groups, and a 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 $\mathsf D (G) -1 < \mathsf d (G, K)$. Thus we disprove the conjecture.
