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Integer‐valued definable functions
Author(s) -
Jones G. O.,
Thomas M. E. M.,
Wilkie A. J.
Publication year - 2012
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bds059
Subject(s) - mathematics , integer (computer science) , infinity , polynomial , exponential function , function (biology) , combinatorics , rational function , exponential polynomial , discrete mathematics , field (mathematics) , pure mathematics , mathematical analysis , evolutionary biology , computer science , biology , programming language
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 definable, analytic function f : [0, ∞) k →ℝ is such that f (ℕ k ) ⊆ ℤ, then either sup |x̄|⩽ r | f (x̄)| grows faster than exp( r δ ), for some δ>0, or f is a polynomial over ℚ.