On the Davenport Constant and Group Algebras

For a finite abelian group G and a splitting field K of G, let d(G, K) denote the largest integer l ∈ N for which there is a sequence S = g1 · . . . · gl over G such that (X g1 − a1) · . . . · (Xl − al) 6= 0 ∈ K[G] for all a1, . . . , al ∈ K . If D(G) denotes the Davenport constant of G, then there is the straightforward inequality D(G)−1 ≤ 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.