Abstract
We study a zero-sum problem dealing with minimal zero-sum sequences of maximal length over finite abelian groups. A positive answer to this problem yields a structural description of sets of lengths with maximal elasticity in transfer Krull monoids over finite abelian groups.
This work was supported by the Austrian Science Fund FWF, Project Numbers W1230 and P33499-N.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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.
F. Chen and S. Savchev, Long minimal zero-sum sequences in the groups \({C}_2^{r-1} \oplus {C}_{2k}\), Integers 14 (2014), Paper A23.
A. Geroldinger and F. Halter-Koch, Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics, vol. 278, Chapman & Hall/CRC, 2006.
A. Geroldinger, M. Liebmann, and A. Philipp, On the Davenport constant and on the structure of extremal sequences, Period. Math. Hung. 64 (2012), 213–225.
A. Geroldinger and I. Ruzsa, Combinatorial Number Theory and Additive Group Theory, Advanced Courses in Mathematics - CRM Barcelona, Birkhäuser, 2009.
A. Geroldinger and R. Schneider, On Davenport’s constant, J. Comb. Theory, Ser. A 61 (1992), 147 – 152.
A. Geroldinger and Q. Zhong, Long sets of lengths with maximal elasticity, Can. J. Math. 70 (2018), 1284–1318.
A. Geroldinger and Q. Zhong, Factorization theory in commutative monoids, Semigroup Forum 100 (2020), 22–51.
B. Girard, An asymptotically tight bound for the Davenport constant, J. Ec. Polytech. Math. 5 (2018), 605–611.
B. Girard and W.A. Schmid, Direct zero-sum problems for certain groups of rank three, J. Number Theory 197 (2019), 297–316.
D.J. Grynkiewicz, Structural Additive Theory, Developments in Mathematics 30, Springer, Cham, 2013.
Chao Liu, On the lower bounds of Davenport constant, J. Comb. Theory, Ser. A 171 (2020), 105162, 15pp.
W.A. Schmid, A realization theorem for sets of lengths, J. Number Theory 129 (2009), 990 – 999.
W.A. Schmid, Some recent results and open problems on sets of lengths of Krull monoids with finite class group, in Multiplicative Ideal Theory and Factorization Theory, Springer, 2016, pp. 323–352.
Acknowledgements
We thank the reviewers for their careful reading.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Bashir, A., Geroldinger, A., Zhong, Q. (2021). On a Zero-Sum Problem Arising From Factorization Theory. In: Nathanson, M.B. (eds) Combinatorial and Additive Number Theory IV. CANT 2020. Springer Proceedings in Mathematics & Statistics, vol 347. Springer, Cham. https://doi.org/10.1007/978-3-030-67996-5_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-67996-5_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-67995-8
Online ISBN: 978-3-030-67996-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)