Year: 2026
Type: preprint
Abstract: The Davenport constant $\DD(G)$ of a finite abelian group $G$ is a classical invariant arising from the arithmetic of ideal class groups of number fields. Closely related to the Davenport constant is the first Narkiewicz constant $\Nark_1(G)$, which arises in the quantitative study of algebraic integers and their factorizations. Gao and Geroldinger conjectured in 2006 that $\DD(G)\leq \sum_{i=1}^r n_i,$ while Narkiewicz and Śliwa conjectured in 1982 that $\Nark_1(G)=\sum_{i=1}^r n_i,$ where $n_1,\ldots,n_r$ are the invariant factors of $G$. Both are central conjectures concerning arithmetic invariants arising from the theory of number fields. In this paper, we disprove both conjectures. Moreover, we show that, for every fixed $r\geq5$, both $\DD(G)-\sum_{i=1}^r n_i$ and $\Nark_1(G)-\sum_{i=1}^r n_i$ are unbounded over finite abelian groups of rank $r$. We develop a probabilistic representation technique to establish a uniform upper bound for the Davenport constant: $$ \DD(G)\leq\frac{16}{5}\,\rr(G)\exp(G). $$ Our bound gives the first improvement of the classical van Emde Boas--Kruyswijk estimate: $\DD(G)\leq\exp(G) \left(1+\log\frac{|G|}{\exp(G)}\right) \leq\exp(G)\bigl(1+(\rr(G)-1)\log\exp(G)\bigr)$, by removing the logarithmic dependence on the exponent from this classical estimate, replacing it with the absolute constant $16/5$. Their bound stood unimproved since its establishment in 1969. The existence of $C=16/5$ also brings us closer to the most challenging conjecture on the Davenport constant, namely the Emde Boas--Kruyswijk Conjecture in 1969, which predicts that $\DD(C_n^r)=r(n-1)+1.$ As a further application of our results, we disprove a conjecture of Girard in 2008 concerning cross numbers, an invariant with deep roots in the arithmetic of number fields and Krull monoids.
Source: arXiv (Cornell University)
Author: Wang, Guoqing
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214381443