Monoids of sequences over finite abelian groups defined via zero-sums with respect to a given set of weights and applications to factorizations of norms of algebraic integers

Abstract

The investigation of the arithmetic of monoids of zero-sum sequences over finite abelian groups is a classical subject due to their crucial role in understanding the arithmetic of (transfer) Krull monoids. More recently, sequences that admit, for a given set of weights, a weighted zero-sum received increased attention. Yet, the focus was on zero-sum constants rather than the arithmetic of the monoids formed by these sequences. We begin a systematic study of the arithmetic of these monoids. We show that for a wide class of weights unions of sets of lengths are intervals and we obtain various results on the elasticity of these monoids. More detailed results are obtained for the special case of plus-minus weighted sequences. Moreover, we apply our results to obtain results on factorizations of norms of algebraic integers.

Keywords:

• Algebraic integer
• finite abelian group
• norm
• weighted subsum
• zero-sum problem

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 11B30
• 11B75
• 11R04
• 20K01

Acknowledgments

The authors thank A. Geroldinger and Q. Zhong for having shared the key-idea of Proposition 3.8 and other suggestions, F. Halter-Koch for an improved version of the proof of Theorem 7.1 and the referee for numerous helpful remarks.

Additional information

Funding

The research of S. Boukheche was supported by a Profas B + fellowship. The research of W. A. Schmid is supported by the project ANR-21-CE39-0009-BARRACUDA.