Nevanlinna Theory and unique range sets in positive characteristic

Abdelbaki Boutabaa and Alain Escassut

Abstract

Let Kbe a complete ultrametric algebraically closed field of characteristic p. We show that Nevanlinna's main Theorem holds, with however some corrections. For instance, we can again characterize all solutions of Yoshida's equation with constant coefficients. Next, we apply the Nevanlinna Theory to find unique range sets (urs) for entire functions in K. For every integer $n\ge3$ we can find urs of n points, except for n=3when p=3. If n and p are relatively prime then any affinely rigid set of n elements is an urs for entire functions, as in characteristic 0.