Plus-minus weighted zero-sum sequences and applications to factorizations of norms of quadratic integers

Abstract: Let $(G,+)$ be a finite abelian group. A sequence $g_1, \dots, g_k$ over $G$ is called a zero-sum sequence if $g_1 + \dots + g_k = 0$ (we consider sequences that just differ by the ordering of the terms as equal). The concatenation of two zero-sum sequences is a zero-sum sequence and the set of all zero-sum sequences over $G$ is thus a monoid. The arithmetic of these monoids has been the subject much investigation.

A sequence is called a \emph{plus-minus weighted zero-sum sequence} if there is a choice of weights $w_i \in \{-1, +1\}$ such that $w_1g_1 + \dots + w_k g_k = 0$. The set of all plus-minus weighted zero-sum sequences over $G$ is a monoid as well. We present some results on the arithmetic of these monoids. Moreover, applications to factorizations of norms of quadratic integers are discussed.

Joint work with S. Boukheche, K. Merito and O. Ordaz.

number theory

Audience: researchers in the topic

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Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

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