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 $u_1,\dots ,u_k,v_1,\dots ,v_m\in H$ such that $u_1\cdot \dots \cdot u_k=v_1\cdot \dots \cdot v_m$. 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 $k^{\ast }\in \mathbb{N}$ such that ${\rho }_{2k+1}(H)=k\mathsf{D}(G)+\lfloor \frac{\mathsf{D}(G)}{2}\rfloor$ for every $k\ge k^{\ast }$. This confirms a conjecture of A. Geroldinger, D. Grynkiewicz, and P. Yuan in \cite{Ge-Gr-Yu15}.

References in corpus (1)

• On products of k atoms II

Cited by in corpus (2)

• Structural properties of subadditive families with applications to factorization theory
• Long sets of lengths with maximal elasticity