A theorem of G. PoÁ lya states that an entire function of exponential order less than 1 or of exponential order 1 and type less then log 2 which takes integer values on the set N of nonnegative integers is a polynomial. The bound log 2 is the best possible since the map z [ 2 Z is not a polynomial and takes integer values on the set N. The aim of this paper is to prove for the polynomial ring F q [T] a theorem similar to the theorem of PoÁ lya.
1997 Academic Press A theorem of PoÁ lya [9], states that an entire function of exponential order less than 1 or of exponential order equal to 1 and of type less then log 2 which takes integer values on the set N of nonnegative integers is a polynomial. The bound log 2 is the best possible since the map z [ 2 Z is not a polynomial and takes integer values on the set of nonnegative integers.
In the following, we prove an analog to the theorem of PoÁ lya for the ring F q [T ] of polynomials over the finite field F q . The statement of this theorem needs some definitions.
I. DEFINITIONS AND NOTATIONS
Let q be a power of a prime number p. We denote by A the ring F q [T ] and by K the field F q (T ). Let &=& denote the infinite valuation on the field K. Then,