Lagrange polynomials over the integers of an imaginary quadratic field
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'fond, Invent. Math. 63, 495-506 (1981; Zbl 0461.10028)] who extended Pólya's theorem to entire functions in \(\mathbb C\) which map \(\mathbb Z[i]\) into \(\mathbb Z[i]\). Further properties of these polynomials and their derivatives have been investigated by \textit{P. Philippon} [Polynômes d'interpolation sur \(\mathbb Z\) et \(\mathbb Z[i]\), Fifty years of polynomials, Proc. Conf. in Honour of Alain Durand, Paris/Fr. 1988, Lect. Notes Math. 1415, 180-195 (1990; Zbl 0701.11023)] Here the authors extend these estimates to any imaginary quadratic number field. As a consequence, they give a variant of Pólya's theorem for integer valued meromorphic functions. In a forthcoming paper, the first named author applies the main estimate to sharpen the measures of linear independence of logarithms of algebraic points on elliptic curves with complex multiplication.
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