Gelfond-Gramain’s Theorem for Function Fields

A theorem of G. POLYA, c.f. [8], states that an entire function of order less than 1 or of order 1 and of type less than log 2 which takes integral values on the set ℕ of natural integers is a polynomial. By interpolation on Gauss integers, A. GEL’FOND, cf. [4], has obtained an analog result for entire functions mapping the ring of Gauss integers in itself. Further, Gel’fond’s theorem was improved. The first improvements always used interpolation polynomials and did not give optimal results. For instance, we quote the work of L. GRUMAN, cf. [6], or the work of D. MASSER, cf. [7]. Using transcendental methods F. Gramain, cf. [5], proved the best possible result for the ring of integers of an imaginary quadratic field. In what follows we give analog results for the ring $$ {\mathbb{F}_{q}}[T] $$ of polynomials over a finite field, and in the case where q is odd, for the ring $$ {\mathbb{F}_{q}}[T,\sqrt {D} ] $$ , integral closure of $$ {\mathbb{F}_{q}}[T] $$ in the quadratic extension $$ {\mathbb{F}_{q}}(T)(\sqrt {D} ) $$ . We do not get the best possible results. However, our method has the merit to provide results, even in the case of real extensions.