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Multi angle On complex continuations of functions definable in $\mathbb{R}_{an,exp}$ with a diophantine application

Auteurs : Wilkie, Alex J. (Auteur de la Conférence)
CIRM (Editeur )

Résumé : Diophantine properties of subsets of $\mathbb{R}^n$ definable in an o-minimal expansion of the ordered field of real numbers have been much studied over the last few years and several applications to purely number theoretic problems have been made. One line of inquiry attempts to characterise the set of definable functions $f : \mathbb{R} \to \mathbb{R}$ having the property that $f(\mathbb{N}) \subset \mathbb{N}$. For example, a result of Thomas, Jones and myself shows that if the structure under consideration is $\mathbb{R}_{exp}$ (the real field expanded by the exponential function) and if, for all positive $r, f(x)$ eventually grows more slowly than $exp(x^r)$, then $f$ is necessarily a polynomial with rational coefficients. In this talk I shall improve this result in two directions. Firstly, I take the structure to be $\mathbb{R}_{an,exp}$ (the expansion of $\mathbb{R}_{exp}$ by all real analytic functions defined on compact balls in $\mathbb{R}^n$) and secondly, I allow the growth rate to be $x^N \cdot 2^x$ for arbitrary (fixed) $N$. The conclusion is that $f(x) = p(x) \cdot 2^x + q(x)$ for sufficiently large $x$, where $p$ and $q$ are polynomials with rational coefficients.

I should mention that over ninety years ago Pólya established the same result for entire functions $f : \mathbb{C} \to \mathbb{C}$ and that in 2007 Langley weakened this assumption to $f$ being regular in a right half-plane of $\mathbb{C}$. I follow Langley’s method, but first we must consider which $\mathbb{R}_{an,exp}$-definable functions actually have complex continuations to a right half-plane and, as it turns out, which of them have a definable such continuation.

Codes MSC :
03C64 - Model theory of ordered structures; o-minimality
 Informations sur la Vidéo Réalisateur : Hennenfent, Guillaume Langue : Anglais Date de publication : 03/03/15 Date de captation : 18/02/15 Collection : Research talks Format : quicktime ; audio/x-aac Durée : 00:58:55 Domaine : Logic and Foundations Audience : Chercheurs ; Doctorants , Post - Doctorants Download : http://videos.cirm-math.fr/2015-02-18_Wilkie.mp4 Informations sur la rencontre Nom du congrès : Real singularities and applications / Singularités réelles et applicationsOrganisteurs Congrès : Dutertre, Nicolas ; Trotman, DavidDates : 16/02/15 - 20/02/15 Année de la rencontre : 2015 URL Congrès : http://chairejeanmorlet-1stsemester2015....Citation DataDOI : 10.24350/CIRM.V.18703403Cite this video as: Wilkie, Alex J. (2015). On complex continuations of functions definable in $\mathbb{R}_{an,exp}$ with a diophantine application. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18703403URI : http://dx.doi.org/10.24350/CIRM.V.18703403