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December 2020 The $N$-measure for algebraic integers having all their conjugates in a sector
Valérie Flammang
Rocky Mountain J. Math. 50(6): 2035-2045 (December 2020). DOI: 10.1216/rmj.2020.50.2035

Abstract

Let α be a nonzero algebraic integer of degree d whose all conjugates α1=α,α2,,αd lie in a sector |argz|𝜃, 0𝜃90. We define the N-measure of α by N(α)=i=1d(|αi|+1|αi|) and the absolute N-measure of α by ν(α)=N(α)1 deg(α). Firstly, we consider the case 𝜃=0. We prove that N(α) and that, if α is a reciprocal algebraic integer, N(α) is a square. Then, we study the set 𝒩 of the quantities ν(α). We prove that there exists a number l such that 𝒩 is dense in (l,). Finally, using the method of auxiliary functions, we find the seven smallest points of 𝒩 in (2,l). In case of 0<𝜃90, we compute the greatest lower bound c(𝜃) of the absolute N-measure of α, for α belonging to eight subintervals of ]

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Valérie Flammang. "The $N$-measure for algebraic integers having all their conjugates in a sector." Rocky Mountain J. Math. 50 (6) 2035 - 2045, December 2020. https://doi.org/10.1216/rmj.2020.50.2035

Information

Received: 2 May 2020; Revised: 31 May 2020; Accepted: 31 May 2020; Published: December 2020
First available in Project Euclid: 5 January 2021

Digital Object Identifier: 10.1216/rmj.2020.50.2035

Subjects:
Primary: 11C08, 11R06, 11Y40

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium

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Vol.50 • No. 6 • December 2020
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