Quelques propriétés des solutions de certaines équations aux différences dans les corps ultramétriques

Abstract

This thesis is devoted to the study of certain difference equations in an ultrametric setting. We focus on the properties of meromorphic solutions defined over a complete field endowed with a non-Archimedean absolute value. First, we introduce the fundamental concepts of ultrametric analysis, including padic numbers and analytic and meromorphic functions. Then, we develop the main tools of ultrametric Nevanlinna theory, which are essential for studying the growth and value distribution of meromorphic functions. Finally, we apply these tools to investigate certain difference and q-difference equations, highlighting conditions on the coefficients that influence the growth, nature, and asymptotic behavior of their solutions.