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Preprint Number 897

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897. A.J. Wilkie
Complex continuations of R_{an,exp}-definable unary functions with a diophantine application
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Submission date: 26 July 2015.

Abstract:

Let F denote the field of germs at +∞ of R_{an,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 F_{poly} and its unique maximal ideal by m_{poly}, our description picks out a subfield R_{poly} of representatives of the residue field of F_{poly} modulo m_{poly}. 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a construction, in considerably greater generality, was already carried out in 1997 by F-V. Kuhlmann and S. Kuhlmann (unpublished, see arXiv:1206.0711v1 [math.LO]) using valuation theoretic methods, but our main aim here is to investigate the complex extensions of the functions under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 R_{poly} consists precisely of those (germs of) R_{an,exp}-definable unary functions that have an R_{an,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.)

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