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H 1 A characterization of class groups via sets of lengths

Auteurs : Geroldinger, Alfred (Auteur de la Conférence)
CIRM (Editeur )

Résumé : Let $H$ be a Krull monoid with finite class group $G$ and suppose that each class contains a prime divisor (rings of integers in algebraic number fields share this property). For each element $a \in H$, its set of lengths $\mathsf L(a)$ consists of all $k \in \mathbb{N} _0$ such that $a$ can be written as a product of $k$ irreducible elements. Sets of lengths of $H$ are finite nonempty subsets of the positive integers, and we consider the system $\mathcal L (H) = \{ \mathsf L (a) \mid a \in H \}$ of all sets of lengths. It is classical that H is factorial if and only if $|G| = 1$, and that $|G| \le 2$ if and only if $|L| = 1$ for each $L \in \mathcal L(H)$ (Carlitz, 1960).

Suppose that $|G| \ge 3$. Then there is an $a \in H$ with $|\mathsf L (a)|>1$, the $m$-fold sumset $\mathsf L(a) + \ldots +\mathsf L(a)$ is contained in $\mathsf L(a^m)$, and hence $|\mathsf L(a^m)| > m$ for every $m \in \mathbb{N}$. The monoid $\mathcal B (G)$ of zero-sum sequences over $G$ is again a Krull monoid of the above type. It is easy to see that $\mathcal L (H) = \mathcal L \big(\mathcal B (G) \big)$, and it is usual to set $\mathcal L (G) := \mathcal L \big( \mathcal B (G) \big)$. In particular, the system of sets of lengths of $H$ depends only on $G$, and it can be studied with methods from additive combinatorics.
The present talk is devoted to the inverse problem whether or not the class group $G$ is determined by the system of sets of lengths. In more technical terms, let $G'$ be a finite abelian group with $|G'| \ge 4$ and $\mathcal L(G) = \mathcal L(G')$. Does it follow that $G$ and $G'$ are isomorphic ?
The answer is positive for groups $G$ having rank at most two $[1]$ and for groups of the form $G = C_{n}^{r}$ with $r \le (n+2)/6$ $[2]$. The proof is based on the characterization of minimal zero-sum sequences of maximal length over groups of rank two, and on the set $\triangle^*(G)$ of minimal distances of $G$ (the latter has been studied by Hamidoune, Plagne, Schmid, and others ; see the talk by Q. Zhong).

Codes MSC :
11R27 - Units and factorization
13A05 - Divisibility
13F05 - Dedekind, Prüfer, Krull and Mori rings and their generalizations
11B30 - Arithmetic combinatorics; higher degree uniformity
20M13 - Arithmetic theory of monoids

 Informations sur la Vidéo Langue : Anglais Date de publication : 06/10/15 Date de captation : 08/09/15 Collection : Research talks ; Combinatorics ; Number Theory Format : quicktime ; audio/x-aac Durée : 00:25:23 Domaine : Combinatorics ; Number Theory Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2015-09-08_Geroldinger.mp4 Informations sur la rencontre Nom de la rencontre : Additive combinatorics in Marseille / Combinatoire additive à MarseilleOrganisateurs de la rencontre : Hennecart, François ; Plagne, Alain ; Szemerédi, EndreDates : 07/09/15 - 11/09/15 Année de la rencontre : 2015 URL Congrès : http://conferences.cirm-math.fr/1107.htmlCitation Data DOI : 10.24350/CIRM.V.18829703 Cite this video as: Geroldinger, Alfred (2015). A characterization of class groups via sets of lengths. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18829703 URI : http://dx.doi.org/10.24350/CIRM.V.18829703

Bibliographie

1. [1] Geroldinger, A., & Schmid, W. A. (2015). A characterization of class groups via sets of lengths. - http://arxiv.org/abs/1503.04679

2. [2] Geroldinger, A., & Zhong, Q. (2015). A characterization of class groups via sets of lengths II. - http://arxiv.org/abs/1506.05223

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