The inverse problem associated to the Davenport constant for \(C_2 \oplus C_2 \oplus C_{2n}\), and applications to the arithmetical characterization of class groups

Wolfgang A. Schmid

The electronic journal of combinatorics, Tome 18 (2011) no. 1

Résumé

The inverse problem associated to the Davenport constant for some finite abelian group is the problem of determining the structure of all minimal zero-sum sequences of maximal length over this group, and more generally of long minimal zero-sum sequences. Results on the maximal multiplicity of an element in a long minimal zero-sum sequence for groups with large exponent are obtained. For groups of the form $C_2^{r-1}\oplus C_{2n}$ the results are optimal up to an absolute constant. And, the inverse problem, for sequences of maximal length, is solved completely for groups of the form $C_2^2 \oplus C_{2n}$. Some applications of this latter result are presented. In particular, a characterization, via the system of sets of lengths, of the class group of rings of algebraic integers is obtained for certain types of groups, including $C_2^2 \oplus C_{2n}$ and $C_3 \oplus C_{3n}$; and the Davenport constants of groups of the form $C_4^2 \oplus C_{4n}$ and $C_6^2 \oplus C_{6n}$ are determined.

Détail

DOI : 10.37236/520

Classification : 11P70, 11B30
Mots-clés : additive combinatorics, Davenport's constant, inverse theorems, factorization