Monatshefte für Mathematik

, Volume 181, Issue 4, pp 779–795 | Cite as

Products of k atoms in Krull monoids

Article

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 [11] 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 reasonable 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 [13].

Keywords

Non-unique factorizations Sets of lengths Krull monoids Zero-sum sequences 

Mathematics Subject Classification

11B30 11R27 13A05 13F05 20M13 

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Copyright information

© Springer-Verlag Wien 2016

Authors and Affiliations

  1. 1.Mathematical CollegeChina University of GeosciencesBeijingPeople’s Republic of China
  2. 2.Institute for Mathematics and Scientific ComputingUniversity of GrazGrazAustria

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