On the inversion of exponential sums in the neighbourhood of transcendental critical points (Q1059732)

Exponential sums are functions with the representation \[ f(z):=\sum^{n}_{k=0}P_ k(z)e^{a_ kz},\quad n\in {\mathbb{N}}_ 0 \] (0\(\not\equiv P_ k\in {\mathbb{C}}[z]\), \(a_ k\) distinct complex numbers). We present a constructive method to obtain explicitly analytical representations for the inverse function of exponential sums in suitable (winded) neighbourhoods of transcendental critical points, as perhaps \(w=\infty.\) We introduce a new ''local uniformizer'' of type (1/a)lg(w/c) for the considered transcendental singularity of the inverse function. The coefficient a can be determined explicitly by the indicatordiagram of Pólya \((:=convex\) hull of the set \(\{\) \(\bar a_ 0,...,\bar a_ n\}\) [see \textit{R. P. Boas} (ed.): G. Pólya. Collected papers. Vol. II, 198- 203 (1974; Zbl 0319.01021)]. The number c is the highest coefficient of a certain polynomial \(P_ k(z)\). The inverse function of f(z) itself has on a certain Riemann surface (which can be specified explicitly too) a representation as a Dirichlet series in the local uniformizer \(t=(1/a)lg(w/c)\).