Publikation: Integer-valued definable functions
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2012
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Bulletin of the London Mathematical Society. 2012, 44(6), pp. 1285-1291. ISSN 0024-6093. eISSN 1469-2120. Available under: doi: 10.1112/blms/bds059
Zusammenfassung
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|¯x| <= r |f(¯x)| grows faster
than exp(rδ), for some δ > 0, or f is a polynomial over Q.
Zusammenfassung in einer weiteren Sprache
Fachgebiet (DDC)
510 Mathematik
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Reell-analytische Funktionen, O-Minimalität
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JONES, Gareth O., Margaret E. M. THOMAS, Alex J. WILKIE, 2012. Integer-valued definable functions. In: Bulletin of the London Mathematical Society. 2012, 44(6), pp. 1285-1291. ISSN 0024-6093. eISSN 1469-2120. Available under: doi: 10.1112/blms/bds059BibTex
@article{Jones2012Integ-20448,
year={2012},
doi={10.1112/blms/bds059},
title={Integer-valued definable functions},
number={6},
volume={44},
issn={0024-6093},
journal={Bulletin of the London Mathematical Society},
pages={1285--1291},
author={Jones, Gareth O. and Thomas, Margaret E. M. and Wilkie, Alex J.}
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<dcterms:abstract xml:lang="eng">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,∞)<sup>k</sup> → |R is such that f(|N^k) ⊆ |Z, then either sup<sub>|¯x| <= r</sub> |f(¯x)| grows faster<br />than exp(r<sup>δ</sup>), for some δ > 0, or f is a polynomial over Q.</dcterms:abstract>
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