tailieunhanh - Báo cáo toán học: "The inverse problem associated to the Davenport constant for C2 ⊕ C2 ⊕ C2n, and applications to the arithmetical characterization of class groups"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: The inverse problem associated to the Davenport constant for C2 ⊕ C2 ⊕ C2n, and applications to the arithmetical characterization of class groups. | The inverse problem associated to the Davenport constant for C2 T C2 T C2n and applications to the arithmetical characterization of class groups Wolfgang A. Schmid Institute of Mathematics and Scientific Computing University of Graz Heinrichstrafie 36 8010 Graz Austria Submitted Nov 16 2009 Accepted Jan 29 2011 Published Feb 14 2011 Mathematics Subject Classification 11B30 20M13 Abstract 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 cr-1 T c2n 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 c2 T c2n. 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 c2 T c2n and C3 T GV and the Davenport constants of groups of the form c2 T c4n and c2 T c6n are determined. Keywords Davenport constant zero-sum sequence zero-sumfree sequence inverse problem non-unique factorization Krull monoid class group 1 Introduction Let G be an additive finite abelian group. The Davenport constant of G denoted D G can be defined as the maximal length of a minimal zero-sum sequence over G that is the largest f such that there exists a sequence g1. g with g-ì E G such that i 1 gi 0 and Supported by the FWF Project number P18779-N13 . THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P33 1 2iei 9i 0 for each 0 I Q 1 . . Another common way to define this constant is via zero-sum free sequences . one defines d G as the maximal length of a .