Sequences of fields with many solutions to the unit equation
type: Item
instance of: scholarly article
title: Sequences of fields with many solutions to the unit equation
zbMATH Open document ID: 0877.11021
author: David Grant
published in: Rocky Mountain Journal of Mathematics
publication date: 1997-07-27T00:00:00Z
full work available at URL: contents.htm
review text: Let \(K\) be an algebraic number field of degree \(\delta\), unit rank \(r\), with unit group \(U\), and denote by \(N\) the number of solutions of the unit equation \( x+y=1 (x,y\in U)\). \textit{J.-H. Evertse} [Invent. Math. 75, 561-584 (1984; Zbl 0521.10015)] has shown that \(N\leq 3\cdot 7^{\delta +2r}\). On the other hand \textit{T. Nagell} [Ark. Mat. 8, 115-127 (1970; Zbl 0213.06902)]\ proved that for \(\delta\geq 5\) there are infinitely many number fields \(K\) such that \(N\geq 6(2\delta -3)\). The purpose of the present paper is to fill the gap between the above mentioned upper and lower bounds. The author constructs sequences of fields for which \(N\) grows quadratically with \(\delta\).
Mathematics Subject Classification ID: 11D61
zbMATH DE Number: 1026400
zbMATH Keywords: unit equation
zbMATH Keywords: exceptional units
reviewed by: István Gaál
MaRDI profile type: Publication
OpenAlex ID: W1984315518
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DOI: 10.1216/RMJM/1181072034
description: scientific article; zbMATH DE number 1026400