Exponential functions in any characteristics
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.
scientific article; zbMATH DE number 1130462
scientific article