The integer transfinite diameter of intervals and totally real algebraic integers
type: Item
instance of: scholarly article
title: The integer transfinite diameter of intervals and totally real algebraic integers
zbMATH Open document ID: 0892.11033
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1998-03-25T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93675
full work available at URL: jtnb9-1.html
review text: Given an interval \(I\) of the real line, the integer transfinite diameter of \(I\) is defined by \(t_{\mathbb Z}(I) = \inf_P \| P\| _\infty^{1/\partial P}\), where \(\| P\| _\infty\) is the maximum of \(| P(x)| \) on \(I\) and the infimum is over all non-constant \(P\) with integer coefficients. As usual \(\partial P\) denotes the degree of \(P\). The classical transfinite diameter \(t(I)\) has the same definition except that the infimum is over all polynomials with complex coefficients. It is known that \(t(I) = | I| /4\) and for intervals of length greater than \(4\) that \(t_{\mathbb Z}(I) = t(I)\). On the other hand the exact value of \(t_{\mathbb Z}(I)\) is not known for any interval of length less than \(4\). This interesting paper studies \(t_{\mathbb Z}(I)\) for small intervals with one fixed rational endpoint. The authors define and study certain functions \(g_-,g, g_+\) which give lower and upper bounds for \(t_{\mathbb Z}(I)\) for Farey intervals \([p/q,r/s]\) with \(qr-ps = 1\). These are used to prove good upper and lower bounds for \(t_{\mathbb Z}(I)\) for intervals of the form \([r/s,r/s + \delta]\) or \([r/s - \delta,r/s]\), where \(r/s\) is fixed and \(\delta \to 0\). These results generalize those of \textit{F. Amoroso} [Ann. Inst. Fourier 40, 885--911 (1990; Zbl 0713.41004)] and \textit{P. Borwein} and \textit{T. Erdélyi} [Math. Comput. 68, 661--681 (1996; Zbl 0859.11044)]. Extending results of the latter authors, they define critical polynomials \(P\) and critical values \(c_P(I)\) for an interval \(I\) which have the property that if \(Q \in \mathbb Z[x]\) has \(\| Q\| _\infty^{1/\partial Q} < c_P\) then \(P^k\) divides \(Q\), where \(k \geq \gamma \partial Q\), \(\gamma\) being a constant depending only on \(P\) and \(\| Q\| _\infty\). They determine 10 critical polynomials for \([0,1]\), extending results of \textit{E. Aparicio} [J. Approximation Theory 55, 270--278 (1988; Zbl 0663.41008)] and Borwein and Erdélyi (loc. cit.). Finally they obt…
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 11R80
zbMATH DE Number: 1132891
zbMATH Keywords: polynomial
zbMATH Keywords: Mahler measure
zbMATH Keywords: inequality
zbMATH Keywords: integer transfinite diameter
zbMATH Keywords: small intervals
zbMATH Keywords: Farey intervals
zbMATH Keywords: critical polynomials
zbMATH Keywords: totally real algebraic integers
author: Valérie Flammang
author: Georges Rhin
author: Chris Smyth
reviewed by: David W. Boyd
MaRDI profile type: Publication
OpenAlex ID: W2017853853
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DOI: 10.5802/JTNB.193
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description: scientific article; zbMATH DE number 1132891