Let be a Krull monoid with finite class group such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree of is the smallest integer with the following property: for each and each two factorizations of , there exist factorizations of such that, for each , arises from by replacing at most atoms from by at most new atoms. Under a very mild condition on the Davenport constant of , we establish a new and simple characterization of the catenary degree. This characterization gives a new structural understanding of the catenary degree. In particular, it clarifies the relationship between and the set of distances of and opens the way towards obtaining more detailed results on the catenary degree. As first applications, we give a new upper bound on and characterize when .
Soit un monoïde de Krull de groupe de classes fini. On suppose que chaque classe contient un diviseur premier (par exemple, l’anneau des entiers d’un corps de nombres ou l’anneau d’holomorphie d’un corps de fonctions). Le degré de chaînage de est le plus petit entier ayant la propriété suivante : pour tout et toute paire de factorisations de l’élément , il existe des factorisations de telles que, pour chaque , on puisse obtenir à partir de en modifiant au plus atomes. Dans cet article, nous obtenons une nouvelle caractérisation du degré de chaînage pour les dont la constante de Davenport du groupe de classes vérifie une certaine hypothèse très peu restrictive. Cette caractérisation offre un nouveau point de vue, plus structurel, sur la notion de degré de chaînage. En particulier, elle clarifie la relation entre et l’ensemble des distances de et permet d’envisager l’obtention de résultats plus précis sur le degré de chaînage. Nous illustrons ce phénomène en donnant deux applications : une nouvelle borne supérieure pour et la caractérisation des tels que .
Keywords: non-unique factorizations, Krull monoids, catenary degree, zero-sum sequence
@article{JTNB_2011__23_1_137_0, author = {Alfred Geroldinger and David J. Grynkiewicz and Wolfgang A. Schmid}, title = {The catenary degree of {Krull} monoids {I}}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {137--169}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {23}, number = {1}, year = {2011}, doi = {10.5802/jtnb.754}, mrnumber = {2780623}, zbl = {1253.11101}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.754/} }
TY - JOUR AU - Alfred Geroldinger AU - David J. Grynkiewicz AU - Wolfgang A. Schmid TI - The catenary degree of Krull monoids I JO - Journal de théorie des nombres de Bordeaux PY - 2011 SP - 137 EP - 169 VL - 23 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.754/ DO - 10.5802/jtnb.754 LA - en ID - JTNB_2011__23_1_137_0 ER -
%0 Journal Article %A Alfred Geroldinger %A David J. Grynkiewicz %A Wolfgang A. Schmid %T The catenary degree of Krull monoids I %J Journal de théorie des nombres de Bordeaux %D 2011 %P 137-169 %V 23 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.754/ %R 10.5802/jtnb.754 %G en %F JTNB_2011__23_1_137_0
Alfred Geroldinger; David J. Grynkiewicz; Wolfgang A. Schmid. The catenary degree of Krull monoids I. Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 1, pp. 137-169. doi : 10.5802/jtnb.754. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.754/
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