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Integer-valued definable functions

Type of Publication: Journal article
URI (citable link): http://nbn-resolving.de/urn:nbn:de:bsz:352-204488
Author: Jones, Gareth O.; Thomas, Margaret; Wilkie, Alex J.
Year of publication: 2012
Published in: Bulletin of the London Mathematical Society ; 44 (2012), 6. - pp. 1285-1291. - ISSN 0024-6093. - eISSN 1469-2120
DOI (citable link): https://dx.doi.org/10.1112/blms/bds059
Summary:
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.
Subject (DDC): 510 Mathematics
Keywords: Reell-analytische Funktionen, O-Minimalität
Link to License: In Copyright
Bibliography of Konstanz: Yes

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JONES, Gareth O., Margaret THOMAS, Alex J. WILKIE, 2012. Integer-valued definable functions. In: Bulletin of the London Mathematical Society. 44(6), pp. 1285-1291. ISSN 0024-6093. eISSN 1469-2120. Available under: doi: 10.1112/blms/bds059

@article{Jones2012Integ-20448, title={Integer-valued definable functions}, year={2012}, doi={10.1112/blms/bds059}, number={6}, volume={44}, issn={0024-6093}, journal={Bulletin of the London Mathematical Society}, pages={1285--1291}, author={Jones, Gareth O. and Thomas, Margaret and Wilkie, Alex J.} }

eng 2013-10-31T23:25:05Z 2012-11-20T09:04:11Z Wilkie, Alex J. Jones, Gareth O. Thomas, Margaret 2012 Thomas, Margaret Integer-valued definable functions terms-of-use Wilkie, Alex J. 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. Jones, Gareth O. The Bulletin of the London Mathematical Society ; 44 (2012), 6. - S. 1285-1291

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