A new exceptional polynomial for the integer transfinite diameter of \([0,1]\)
type: Item
instance of: scholarly article
title: A new exceptional polynomial for the integer transfinite diameter of \([0,1]\)
zbMATH Open document ID: 1071.11019
author: Qiang Wu
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2005-06-30T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93919
review text: The integer Chebyshev polynomials \(P_k\) on \(I=[0,1]\) are defined as polynomials of degree \(k\) with integral coefficients whose maximal absolute value \(M_k\) in \(I\) is minimal among all non-zero polynomials of degree \(k\) in \(Z[X]\). The integer transfinite diameter \(t_Z(I)\) of \(I\) is defined as the limit of \(M_k^{1/k}\) for \(k\to\infty\). A list of the polynomials \(P_k\) for \(k\leq75\) has been given by \textit{L. Habsieger} and \textit{B. Salvy} [Math. Comput. 66, 763--770 (1997; Zbl 0911.11033)], and now the author lists them for \(76\leq k\leq100\). It has been asked by \textit{P. Borwein} and \textit{T. Erdelyi} [Math. Comput. 65, 661--681 (1996; Zbl 0859.11044)] whether all zeros of the \(P_k\)'s lie in \([0,1]\). A negative answer with \(k=70\) was provided in the paper of Habsieger and Salvy quoted above, and now the author shows that \(P_{95}\) also has non-real zeros. Finally the bound \(t_Z(I)<0.423164171\) is established, improving slightly the previous bound \(t_Z(I)<0.4232\), due to I. E. Pritsker (to appear).
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 2184628
zbMATH Keywords: integer Chebyshev polynomials
zbMATH Keywords: transfinite diameter
MaRDI profile type: Publication
OpenAlex ID: W2321530583
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DOI: 10.5802/JTNB.430
description: scientific article; zbMATH DE number 2184628