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2012.1: Integer-valued definable functions

2012.1: A J Wilkie, G O Jones and M E M Thomas (2012) Integer-valued definable functions.

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We present a dichotomy, in terms of growth at infinity, of analytic functions definable in the real exponential field which take integer values at natural number inputs. Using a result concerning the density of rational points on curves definable in this structure, we show that if a function f : [0;1)n ! R is such that f(Nn) Z, then either supjxjr f(x) grows faster than exp(r), for some > 0, or f is a polynomial over Q.

Item Type:MIMS Preprint
Uncontrolled Keywords:O-minimality, number theory, CICADA
Subjects:MSC 2000 > 03 Mathematical logic and foundations
MSC 2000 > 11 Number theory
MSC 2000 > 26 Real functions
MIMS number:2012.1
Deposited By:Prof Alex J Wilkie
Deposited On:05 January 2012

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