Exponential functions in any characteristics
type: Item
instance of: scholarly article
title: Exponential functions in any characteristics
zbMATH Open document ID: 0893.11022
DOI: 10.1007/PL00000047
author: Laurent Denis
published in: Aequationes Mathematicae
publication date: 1998-04-27T00:00:00Z
review text: A generalization of the notion of Drinfeld modules to fields of any characteristic has been introduced by \textit{Y. Hellegouarch} in 1990 [Approximations diophantiennes et nombres transcendants, C.-R. Colloq., Luminy/Fr. 1990, 123-164 (1992; Zbl 0760.11019)]. Here the author introduces new functions which generalize at the same time the Carlitz exponential (in finite characteristic) and the function \(e^{z}-1\) (in zero characteristic). He gives the following nice example of his general setup. There is a unique power series \(E(z)\) in \(\mathbb{Z}((1/T))z\) with \(E'(0)=1\) which satisfies \[ E(Tz)=TE(z)+E(z)^{2}. \] The reduction modulo \(2\) of \(E(z)\) is the Carlitz exponential over \(\mathbb{F}_{2}[T]\), while the specialization at \(T=2\) is the function \(e^{z}-1\). More generally, given a field \(K\) of zero characteristic, the author solves the functional equation \[ f(Tz)=Tf(z)+a_{2}f(z)^{2}+\cdots+a_{d}f(z)^{d} \] when \(a_{2},\ldots,a_{d}\) belong to an algebraically closed field \(C\) containing \(K((1/T))\) (also \(C\) is assumed to be complete with respect to the valuation \(v\) such that \(v(P)=-\deg_{T}P\) for \(P\in K[T]\)). There is a unique solution \(f\) for which \(f'(0)=1\). The author studies the transcendence and algebraic independence of such functions. He also considers connections with Carlitz exponentials as well as with the usual exponential or trigonometric functions.
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11J91
zbMATH DE Number: 1130462
zbMATH Keywords: exponential functions
zbMATH Keywords: Carlitz exponential
zbMATH Keywords: finite characteristic
zbMATH Keywords: functional equation
zbMATH Keywords: transcendence
zbMATH Keywords: algebraic independence
reviewed by: Michel Waldschmidt
MaRDI profile type: Publication
full work available at URL: pl00000047
OpenAlex ID: W2014274257
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description: scientific article; zbMATH DE number 1130462