The catenary degree of Krull monoids I

Geroldinger, Alfred; Grynkiewicz, David J.; Schmid, Wolfgang A.

Journal de théorie des nombres de Bordeaux, Tome 23 (2011) no. 1, pp. 137-169.

Soit $H$ un monoïde de Krull de groupe de classes $G$ 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 $\mathsf{c}\left(H\right)$ de $H$ est le plus petit entier $N$ ayant la propriété suivante : pour tout $a\in H$ et toute paire de factorisations $z,z^{\prime }$ de l’élément $a$, il existe des factorisations $z=z_0,...,z_k=z^{\prime }$ de $a$ telles que, pour chaque $i\in [1,k]$, on puisse obtenir $z_i$ à partir de $z_{i-1}$ en modifiant au plus $N$ atomes. Dans cet article, nous obtenons une nouvelle caractérisation du degré de chaînage pour les $H$ 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 $\mathsf{c}\left(H\right)$ et l’ensemble des distances de $H$ 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 $\mathsf{c}\left(H\right)$ et la caractérisation des $H$ tels que $\mathsf{c}\left(H\right)\le 4$.

Let $H$ be a Krull monoid with finite class group $G$ 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 $\mathsf{c}\left(H\right)$ of $H$ is the smallest integer $N$ with the following property: for each $a\in H$ and each two factorizations $z,z^{\prime }$ of $a$, there exist factorizations $z=z_0,...,z_k=z^{\prime }$ of $a$ such that, for each $i\in [1,k]$, $z_i$ arises from $z_{i-1}$ by replacing at most $N$ atoms from $z_{i-1}$ by at most $N$ new atoms. Under a very mild condition on the Davenport constant of $G$, 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 $\mathsf{c}\left(H\right)$ and the set of distances of $H$ and opens the way towards obtaining more detailed results on the catenary degree. As first applications, we give a new upper bound on $\mathsf{c}\left(H\right)$ and characterize when $\mathsf{c}\left(H\right)\le 4$.

[1] J. Amos, S.T. Chapman, N. Hine, and J. Paixao, Sets of lengths do not characterize numerical monoids. Integers 7 (2007), Paper A50, 8p. | MR | Zbl

[2] D.D. Anderson, S.T. Chapman, F. Halter-Koch, and M. Zafrullah, Criteria for unique factorization in integral domains. J. Pure Appl. Algebra 127 (1998), 205–218. | MR | Zbl

[3] P. Baginski, S.T. Chapman, R. Rodriguez, G.J. Schaeffer, and Y. She, On the delta set and catenary degree of Krull monoids with infinite cyclic divisor class group. J. Pure Appl. Algebra 214 (2010), 1334 – 1339. | MR | Zbl

[4] G. Bhowmik and J.-C. Schlage-Puchta, Davenport’s constant for groups of the form ${\mathbb{Z}}_3\oplus {\mathbb{Z}}_3\oplus {\mathbb{Z}}_{3d}$. Additive Combinatorics (A. Granville, M.B. Nathanson, and J. Solymosi, eds.), CRM Proceedings and Lecture Notes, vol. 43, American Mathematical Society, 2007, pp. 307–326. | MR | Zbl

[5] C. Bowles, S.T. Chapman, N. Kaplan, and D. Reiser, On delta sets of numerical monoids. J. Algebra Appl. 5 (2006), 695–718. | MR | Zbl

[6] S.T. Chapman, J. Daigle, R. Hoyer, and N. Kaplan, Delta sets of numerical monoids using nonminimal sets of generators. Commun. Algebra 38 (2010), 2622–2634. | MR

[7] S.T. Chapman, P.A. García-Sánchez, and D. Llena, The catenary and tame degree of numerical monoids. Forum Math. 21 (2009), 117 – 129. | MR | Zbl

[8] S.T. Chapman, P.A. García-Sánchez, D. Llena, and J. Marshall, Elements in a numerical semigroup with factorizations of the same length. Can. Math. Bull. 54 (2010), 39–43.

[9] S.T. Chapman, P.A. García-Sánchez, D. Llena, V. Ponomarenko, and J.C. Rosales, The catenary and tame degree in finitely generated commutative cancellative monoids. Manuscr. Math. 120 (2006), 253–264. | MR | Zbl

[10] S.T. Chapman, R. Hoyer, and N. Kaplan, Delta sets of numerical monoids are eventually periodic. Aequationes Math. 77 (2009), 273–279. | MR | Zbl

[11] Y. Edel, Sequences in abelian groups $G$ of odd order without zero-sum subsequences of length $\exp \left(G\right)$. Des. Codes Cryptography 47 (2008), 125–134. | MR | Zbl

[12] Y. Edel, C. Elsholtz, A. Geroldinger, S. Kubertin, and L. Rackham, Zero-sum problems in finite abelian groups and affine caps. Quarterly. J. Math., Oxford II. Ser. 58 (2007), 159–186. | MR | Zbl

[13] Y. Edel, S. Ferret, I. Landjev, and L. Storme, The classification of the largest caps in $AG(5,3)$. J. Comb. Theory, Ser. A 99 (2002), 95 –110. | MR | Zbl

[14] M. Freeze and W.A. Schmid, Remarks on a generalization of the Davenport constant. Discrete Math. 310 (2010), 3373–3389. | MR | Zbl

[15] W. Gao and A. Geroldinger, On long minimal zero sequences in finite abelian groups. Period. Math. Hung. 38 (1999), 179–211. | MR | Zbl

[16] —, Zero-sum problems in finite abelian groups : a survey. Expo. Math. 24 (2006), 337–369. | MR | Zbl

[17] A. Geroldinger, Additive group theory and non-unique factorizations. Combinatorial Number Theory and Additive Group Theory (A. Geroldinger and I. Ruzsa, eds.), Advanced Courses in Mathematics CRM Barcelona, Birkhäuser, 2009, pp. 1–86. | MR | Zbl

[18] A. Geroldinger, D.J. Grynkiewicz, and W.A. Schmid, The catenary degree of Krull monoids II. manuscript.

[19] A. Geroldinger and F. Halter-Koch, Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory. Pure and Applied Mathematics, vol. 278, Chapman & Hall/CRC, 2006. | MR | Zbl

[20] A. Geroldinger and W. Hassler, Arithmetic of Mori domains and monoids. J. Algebra 319 (2008), 3419–3463. | MR | Zbl

[21] A. Geroldinger and J. Kaczorowski, Analytic and arithmetic theory of semigroups with divisor theory. J. Théor. Nombres Bordx. 4 (1992), 199–238. | EuDML | Numdam | MR | Zbl

[22] A. Geroldinger and R. Schneider, On Davenport’s constant. J. Comb. Theory, Ser. A 61 (1992), 147–152. | MR | Zbl

[23] R. Gilmer, Commutative Semigroup Rings. The University of Chicago Press, 1984. | MR | Zbl

[24] P.A. Grillet, Commutative Semigroups. Kluwer Academic Publishers, 2001. | MR | Zbl

[25] F. Halter-Koch, Ideal Systems. An Introduction to Multiplicative Ideal Theory. Marcel Dekker, 1998. | MR | Zbl

[26] A. Iwaszkiewicz-Rudoszanska, On the distribution of coefficients of logarithmic derivatives of $L$-functions attached to certain arithmetical semigroups. Monatsh. Math. 127 (1999), 189–202. | MR | Zbl

[27] —, On the distribution of prime divisors in arithmetical semigroups. Funct. Approximatio, Comment. Math. 27 (1999), 109 – 116. | MR | Zbl

[28] H. Kim, The distribution of prime divisors in Krull monoid domains. J. Pure Appl. Algebra 155 (2001), 203–210. | MR | Zbl

[29] H. Kim and Y. S. Park, Krull domains of generalized power series. J. Algebra 237 (2001), 292–301. | MR | Zbl

[30] C.R. Leedham-Green, The class group of Dedekind domains. Trans. Am. Math. Soc. 163 (1972), 493–500. | MR | Zbl

[31] M. Omidali, The catenary and tame degree of numerical monoids generated by generalized arithmetic sequences. Forum Math., to appear. | MR | Zbl

[32] O. Ordaz, A. Philipp, I. Santos, and W.A. Schmid, On the Olson and the strong Davenport constants. J. Théor. Nombres Bordx., to appear. | EuDML | Numdam | Zbl

[33] A. Potechin, Maximal caps in AG $(6,3)$. Des. Codes Cryptography 46 (2008), 243–259. | MR | Zbl

[34] J.C. Rosales and P.A. García-Sánchez, Numerical Semigroups. Springer, 2009. | MR | Zbl

[35] W.A. Schmid, A realization theorem for sets of lengths. J. Number Theory 129 (2009), 990–999. | MR | Zbl