Applications of the p -adic Nevanlinna theory to functional equations

Let K be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. We apply the p -adic Nevanlinna theory to functional equations of the form g = R ∘ f , where R ∈ K ( x ) , f , g are meromorphic functions in K , or in an “open disk”, g satisfying conditions on the order of its zeros and poles. In various cases we show that f and g must be constant when they are meromorphic in all K , or they must be quotients of bounded functions when they are meromorphic in an “open disk”. In particular, we have an easy way to obtain again Picard-Berkovich’s theorem for curves of genus 1 and 2 . These results apply to equations f m + g n = 1 , when f , g are meromorphic functions, or entire functions in K or analytic functions in an “open disk”. We finally apply the method to Yoshida’s equation y ′ m = F ( y ) , when F ∈ K ( X ) , and we describe the only case where solutions exist: F must be a polynomial of the form A ( y - a ) d where m - d divides m , and then the solutions are the functions of the form f ( x ) = a + λ ( x - α ) m m - d , with λ m - d ( m m - d ) m = A .