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

Safia Boukheche (1, 2) , Kamil Merito (1) , Oscar Ordaz (3) , Wolfgang Schmid (4)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 USTHB - Université des Sciences et de la Technologie Houari Boumediene = University of Sciences and Technology Houari Boumediene [Alger]
3 UCV - Universidad Central de Venezuela
4 LAGA - Laboratoire Analyse, Géométrie et Applications

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.

Mots clés: finite abelian group; weighted subsum; zero-sum problem; algebraic integer; norm

Domaines: Mathématiques [math]

Safia Boukheche, Kamil Merito, Oscar Ordaz, Wolfgang Schmid. 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. Communications in Algebra, 2022, 50 (10), pp.4195-4217. ⟨10.1080/00927872.2022.2058009⟩. ⟨hal-03551892⟩