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].
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Communicated by A. Constantin.
Main part of this manuscript was written while the first author visited University of Graz. She would like to gratefully acknowledge the kind hospitality from the host institute and the authors would like to thank Professor Alfred Geroldinger for his many helpful suggestions. This work was supported by the Austrian Science Fund FWF (Project No. M1641-N26), NSFC (Grant No. 11401542), the Fundamental Research Funds for the Central Universities (Grant No. 2652014033), and the China Scholarship Council.
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Fan, Y., Zhong, Q. Products of k atoms in Krull monoids. Monatsh Math 181, 779–795 (2016). https://doi.org/10.1007/s00605-016-0942-9
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DOI: https://doi.org/10.1007/s00605-016-0942-9