Functional equations in a p-adic context

Recently, in the complex context, several results were obtained concerning functional equations of the form P ( f ) = Q ( g ) where P and Q are polynomials of only two or three terms whose coefficients are small functions: in certain cases the equation does not admit any pair of admissible solutions and in other cases it only admits pairs of solutions that are of a very particular type. Here we consider similar questions when the ground field is a p-adic complete algebraically closed field of characteristic 0 and we derive results that are often analogous. For instance, if f n + a 1 f n − m + b 1 = c ( g − n + a 2 g n − m + b 2 ) , with a i , b j small functions with regard to f, g and a 2 b 2 non-identically 0, then c = b 1 b 2 and f = h g with h m = a 1 a 2 . However, contrary to the complex context, here results apply not only to meromorphic functions defined in the whole field but also to unbounded meromorphic functions defined inside an open disc. The main tool is the p-adic Nevanlinna theory.