Integer valued functions and Carlitz module
type: Item
instance of: scholarly article
title: Integer valued functions and Carlitz module
zbMATH Open document ID: 1235.13015
author: David Adam
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2011-03-07T00:00:00Z
full work available at URL: https://eudml.org/doc/116403
review text: To begin with, recall Gel'fond's theorem [\textit{A. O. Gel'fond}, Mat. Sb. 40, 42--47 (1933; JFM 59.1039.01; Zbl 0007.12102)]: Let \(r\in\mathbb N\setminus \{0,1\}\). An entire function \(f\) on \(\mathbb C\), of quadratic type \(<1/4r\) and such that \(f(r^n)\in\mathbb Z\), for any \(n\in\mathbb N\), is a polynomial. Moreover, the bound \(1/4r\) is optimal. After getting a first analogue of Gel'fond's theorem in [``Car-Pólya and Gel'fond's theorems for \({\mathbb F}_q [T]\)'', Acta Arith. 115, No. 3, 287--303 (2004; Zbl 1088.11087)], the author gives in the paper under review another analogue of Gel'fond's theorem for the finite characteristic case. In order to state this theorem, some definition and notation are needed. Let \(q:=p^n\) be some power of a prime number \(p\) and let \(\Omega\) be the completion of an algebraic closure of \(\mathbb F_q((1/T))\) for the \(1/T\)-adic valuation \(v\) normalized by \(v(T)=-1\). Let \( C\) be the Carlitz module and \(\tau\) the \(q\)-Frobenius. For \(a\in\mathbb F_q[T]\), set \(C_a:=C(a)\). At last, let \(H\in\mathbb F_q[T]\) be a fixed polynomial, with \(h:=\deg(H)\geq 1\) and define \(C_a(H):=C_a(\tau)(H)\in\mathbb F_q(T)\) and \(\mathfrak S:=\{C_a(H)\mid a\in\mathbb F_q[T]\}\). The author proves the following theorem: Let \(H\in\mathbb F_q[T]\), with \(h:=\deg(H)\) be such that \(q^h\geq 3\). An entire function \(f\) over \(\Omega\) of quadratic type \(<1/4h\) such that \(f(C_a(H))\in\mathbb F_q[T]\) for any \(a\in\mathbb F_q[T]\) is a polynomial in \(\mathbb F_q(T)[X]\). Moreover, the bound \(1/4h\) is optimal. This theorem is gotten by building some regular basis of the \(\mathbb F_q[T]\)-module of integer valued polynomials over \(\mathfrak S\).
reviewed by: Martine Picavet-L'Hermitte
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 5862099
zbMATH Keywords: integer valued functions
zbMATH Keywords: Carlitz module
zbMATH Keywords: entire function
zbMATH Keywords: regular basis
MaRDI profile type: Publication
OpenAlex ID: W1966411493
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DOI: 10.5802/JTNB.715
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description: scientific article; zbMATH DE number 5862099