Functional equations in a p-adic context

Abstract

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 = hg 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.

Uniqueness Results on Differential Polynomials Generated by a Meromorphic Function and a L-Function

“…Many mathematicians around the world have found the uniqueness problem of meromorphic functions in the field C or in the p-adic field to be an intriguing research topic and we can see significant number of research works regarding the polynomial of uniqueness in C (see [6], [8], [14], [16], [30], [31]) as well as in p-adic fields (see [1], [4], [5], [19], [20], [21], [22]). After studying the uniqueness problems of the form P (f ) = P (g), where P is a polynomial, the studies were extended to problems of the form f ′ P ′ (f ) = g ′ P ′ (g), in both complex and p-adic contexts.…”

On non-Archimedean recurrence equations and their applications

“…Moreover, the proved Theorem 3.4 allows to find solutions of functional equations, when one takes instead of A the algebra of analytic functions. In [9] polynomial functional equations have been investigated over p-adic analytic functions.…”

On Functional Equations of the Fermat-Waring Type for Non-Archimedean Vectorial Entire Functions

“…In recent years the problem of existence or non-existence of solutions to the equation has investigated by many authors (see [1], [2], [6], [8], [10], [20], [21], [22], [23]). For the case of entire functions of one variable in a non-Archimedean field, many interesting results are obtained ( [4], [5], [6], [9], [11], [12], [13], [16], [17]). …”