Lagrange polynomials over the integers of an imaginary quadratic field
type: Item
instance of: scholarly article
title: Lagrange polynomials over the integers of an imaginary quadratic field
zbMATH Open document ID: 0914.11009
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1999-06-06T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/225312
full work available at URL: jtnb10-1.html
review text: Pólya's proof of his well known result on integer valued entire functions [\textit{G. Pólya}, Über ganzwertige ganze Funktionen, Rend. Circ. Mat. Palermo 40, 1-16 (1915; JFM 45.0665.01)] involves binomial polynomials \(z(z-1)\cdots(z-n+1)/n!\). The first idea of eliminating the factorials from the derivatives of auxiliary functions with the help of such polynomials is due to \textit{N. I. Fel'dman} [On the measure of transcendence of \(\pi\), Am. Math. Soc., Transl., II. Ser. 58, 110-124 (1966); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 24, 357-368 (1960; Zbl 0199.09601) and Approximation of the logarithms of algebraic numbers by algebraic numbers, Am. Math. Soc., Transl., II. Ser. 58, 125-142 (1966); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 24, 475-492 (1960; Zbl 0199.09602)]. These so-called Fel'dman polynomials were one of his key tools when he achieved a best possible dependence for the estimate in terms of the heights of the coefficients for linear independence measures of logarithms of algebraic numbers [\textit{N. I. Fel'dman}, An improvement of the estimate of a linear form in the logarithms of algebraic numbers, Math. USSR, Sb. 6, 393-406 (1968); translation from Mat. Sb., Nov. Ser. 77(119), 423-436 (1968; Zbl 0235.10018)]. Variants of these polynomials and their derivatives have been introduced by A. Baker and E. M. Matveev, who consider the values at rational points only. Lagrange polynomials related to the ring of Gaussian integers \(\mathbb Z[i]\) have been introduced by \textit{L. Gruman }[Propriétés arithmétiques des fonctions entières, Bull. Soc. Math. Fr. 108, 421-440 (1980; Zbl 0466.30020)] (see also \textit{D. Masser} [Sur les fonctions entières à valeurs entières, C. R. Acad. Sci., Paris, Sér. A 291, 1-4 (1980; Zbl 0442.30021)]) and used by \textit{F. Gramain} [Polynômes d'interpolation sur \(\mathbb Z[i]\), Groupe Étude Anal. Ultramétrique, 6è Année 1978/79, Exposé No. 16 (1980; Zbl 0442.30022) and Sur le théorème de Fukasawa-Gel'…
reviewed by: Michel Waldschmidt
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 30D15
Mathematics Subject Classification ID: 11J72
zbMATH DE Number: 1229649
zbMATH Keywords: JFM 45.0665.01
author: Mohammed Ably
author: M. M'zari
MaRDI profile type: Publication
OpenAlex ID: W2089332394
cites work: Q5521594
cites work: Q4071794
cites work: IMPROVED ESTIMATE FOR A LINEAR FORM OF THE LOGARITHMS OF ALGEBRAIC NUMBERS
cites work: Q3884374
cites work: Sur le théorème de Fukasawa-Gel'fond
cites work: Propriétés arithmétiques des fonctions entières
cites work: Q3478466
cites work: Approximate formulas for some functions of prime numbers
cites work: Pólya's theorem by Schneider's method
DOI: 10.5802/JTNB.220
description: scientific article; zbMATH DE number 1229649