A Generalized Davenport Constant of the Second Kind

In this paper, we explore a ring invariant which is closely related to the Davenport constant of a group. In particular, we will calculate this invariant for a certain class of rings of integers and their orders and use it to understand factorization properties of the latter. To this end, we also examine the well-behaved class of Galois-invariant orders.

💡 Research Summary

The paper introduces a novel invariant for orders in rings of integers, denoted (\overline{D}(O)) and called the “generalized Davenport constant of the second kind.” The authors begin by recalling the classical Davenport constant (D(G)) for a finite abelian group (G), emphasizing its two equivalent definitions: (i) the maximal length of a zero‑sum sequence without a non‑empty zero‑sum subsequence, and (ii) the minimal length such that any sequence of that length must contain a non‑empty zero‑sum subsequence. They note the well‑known connection between (D(G)) and factorization in number fields: if (R) is the ring of integers of a number field with class group (\mathrm{Cl}(R)), then the elasticity (\rho(R)) satisfies (\rho(R)=D(\mathrm{Cl}(R))^{2}).

Motivated by the desire to study factorization in non‑Dedekind orders, the authors define, for an atomic monoid (R) and a non‑empty subset (O\subseteq R), an “(O)-product” as a product of irreducibles of (R) that lies in (O). The generalized Davenport constant (D_{O}(R)) is then the smallest integer (n) such that every (R)-product of length (n) contains an (O)-subproduct. When (O) is a domain and (R) its integral closure, they write (\overline{D}(O)=D_{O}(R)). This definition mirrors the second definition of the classical Davenport constant, replacing the zero element by the subset (O).

The paper contrasts (D_{O}(R)) with the (\omega)-invariant, which measures the minimal number of factors needed to guarantee a proper subproduct divisible by a given element. Example 1.8 shows that for the ideal (5\mathbb{Z}