Integer valued functions and Carlitz module
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 \(\mathfrak S\).
scientific article; zbMATH DE number 5862099
scientific article