On the smallest number of terms of vanishing sums of units in number fields

Cs Bertók, K. Győry, L. Hajdu, A. Schinzel

Research output: Contribution to journalArticle

Abstract

Let K be a number field. In the terminology of Nagell a unit ε of K is called exceptional if 1−ε is also a unit. The existence of such a unit is equivalent to the fact that the unit equation ε123=0 is solvable in units ε123 of K. Numerous number fields have exceptional units. They have been investigated by many authors, and they have important applications. In this paper we deal with a generalization of exceptional units. We are interested in the smallest integer k with k≥3, denoted by ℓ(K), such that the unit equation ε1+…+εk=0 is solvable in units ε1,…,εk of K. If no such k exists, we set ℓ(K)=∞. Apart from trivial cases when ℓ(K)=∞ we give an explicit upper bound for ℓ(K). We obtain several results for ℓ(K) in number fields of degree at most 4, cyclotomic fields and general number fields of given degree. We prove various properties of ℓ(K), including its magnitude, parity as well as the cardinality of number fields K with given degree and given odd resp. even value ℓ(K). Finally, as an application, we deal with certain arithmetic graphs, namely we consider the representability of cycles. We conclude the paper by listing some problems and open questions.

Original languageEnglish
Pages (from-to)328-347
Number of pages20
JournalJournal of Number Theory
Volume192
DOIs
Publication statusPublished - Nov 2018

Keywords

  • Arithmetic graphs
  • Exceptional units
  • Unit equations

ASJC Scopus subject areas

  • Algebra and Number Theory

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