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

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 (‘The elementary theory of restricted analytic fields with exponentiation’, Ann. of Math. 140 (1994) 183–205; ‘Logarithmic-exponential power series’, J. London Math. Soc. (2) 56 (1997) 417–434), 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such a construction, in considerably greater generality, was already carried out in 1997 by Kuhlmann and Kuhlmann (unpublished; see arXiv:1206.0711v1 [math.LO]) using valuation-theoretic methods, but our main aim here is to investigate the complex extensions of the functions under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.) We then consider the analogous situation for those (germs of) functions in F having at most exponential growth. We briefly describe the Kuhlmann representation of the residue field and, although the functions therein do not have analytic continuations to a right half-plane in general, they turn out to have very good approximations that do. There has been much research over the last ten years on diophantine properties of sets definable in o-minimal structures (see, for example, L. A. Butler, ‘Some cases of Wilkie's conjecture’, Bull. London Math. Soc. 44 (2012) 642–660; G. O. Jones and M. E. M. Thomas, ‘The density of algebraic points on certain Pfaffian surfaces’, Q. J. Math. 63 (2012) 637–651; G. O. Jones, M. E. M. Thomas and A. J. Wilkie, ‘Integer-valued definable functions’, Bull. London Math. Soc. 44 (2012) 1285–1291; J. Pila and A. J. Wilkie, ‘The rational points of a definable set’, Duke Math. J. 133 (2006) 591–616; A. J. Wilkie, ‘Diophantine properties of sets definable in o-minimal structures’, J. Symbolic Logic 69 (2004) 851–861) and in the final section of this paper we make a small contribution to this work. We apply our results to adapt a method of Pólya (Über ganze ganzwertige Funktionen, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen (1921) 1–10), as modified by Langley (J. K. Langley, ‘Integer-valued analytic functions in a half-plane’, Comput. Methods Funct. Theory 7 (2007) 433–442), and prove the following generalization of a result from G. O. Jones, M. E. M. Thomas and A. J. Wilkie, ‘Integer-valued definable functions’, Bull. London Math. Soc. 44 (2012) 1285–1291: let f : R → R be an R an , exp -definable function such that, for some r ∈ R with, | f ( x ) | ⩽ 2 r x for all sufficiently large x. Assume also that f ( n ) ∈ Z for all sufficiently large n ∈ N . Then there exists a polynomial P with rational coefficients such that f ( x ) = P ( x ) for all sufficiently large x.