Products of $k$ atoms in Krull monoids

Abstract

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor. For $k\in \mathbb N$, let $\mathcal U_k(H)$ denote the set of all $m\in \mathbb N$ with the following property: There exist atoms $u_1, \ldots, u_k, v_1, \ldots , v_m\in H$ such that $u_1\cdot\ldots\cdot u_k=v_1\cdot\ldots\cdot v_m$. It is well-known that the sets $\mathcal U_k (H)$ are finite intervals whose maxima $\rho_k(H)=\max \mathcal U_k(H) $ depend only on $G$. If $|G|\le 2$, then $\rho_k (H) = k$ for every $k \in \mathbb N$. Suppose that $|G| \ge 3$. An elementary counting argument shows that $\rho_{2k}(H)=k\mathsf D(G)$ and $k\mathsf D(G)+1\le \rho_{2k+1}(H)\le k\mathsf D(G)+\lfloor \frac{\mathsf D(G)}{2}\rfloor$ where $\mathsf D(G)$ is the Davenport constant. In \cite{Ga-Ge09b} it was proved that for cyclic groups we have $k\mathsf D(G)+1 = \rho_{2k+1}(H)$ for every $k \in \mathbb N$. In the present paper we show that (under a mild condition on the Davenport constant) for every noncyclic group there exists a $k^*\in \mathbb N$ such that $\rho_{2k+1}(H)= k\mathsf D(G)+\lfloor \frac{\mathsf D(G)}{2}\rfloor$ for every $k\ge k^*$. This confirms a conjecture of A. Geroldinger, D. Grynkiewicz, and P. Yuan in \cite{Ge-Gr-Yu15}.