Complex continuations of ℝan, exp-definable unary functions with a diophantine application

Let F denote the field of germs at +∞ of Ran,exp-definable unary functions. Starting from its characterization in terms of closure conditions as given by van den Dries, Macintyre and Marker ([3], [4]), we give a similar description of its subring consisting of the germs of polynomial growth. More precisely, denoting this ring by Fpoly and its unique maximal ideal by mpoly, our description picks out a subfield Rpoly of representatives of the residue field of Fpoly modulo mpoly. In fact, such a construction, in considerably greater generality, was already carried out in 1997 by F-V. Kuhlmann and S. Kuhlmann (unpublished, see arXiv:1206.0711v1 [math.LO]) using valuation theoretic methods, but our main aim here is to investigate the complex extensions of the functions under consideration. It turns out that Rpoly consists precisely of those (germs of) Ran,exp-definable unary functions that have an Ran,exp-definable analytic continuation to a right half plane of C and we use this fact to give a different proof of the Kuhlmann result. (Roughly speaking, the (real) valuation theory is replaced by the Phragmén-Lindelöf method applied to the complex continuations.)