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Distribution de valeurs des fonctions méromorphes ultramétriques, application de la théorie de Nevanlinna

Abstract : We study properties of meromorphic functions in a complete ultrametric algebraically closed eld of characteristic zero that we denote K ex: K = Cp and similar properties in an open disk of K, taking into account Lazard's problem, that we avoid considering a spherically complete extension of K. On one hand, the problems studied concern the distribution of zeroes, exceptional values, for various type of ultrametric meromorphic functions in K or inside an open disk of K and particularly Hayman's Conjecture in an ultrametric eld. On the other hand, problems of uniqueness are examined for meromorphic functions in the whole eld K or in an open disk of K satisfying certain hypotheses: functions of the form (P f)0 and (P g)0 where P is a polynomial satisfying certain condition and sharing a small function with respect to f and g, counting multiplicities. This last type of problems show some connections with questions on polynomial of uniqueness for meromorphic functions and with questions on Unique Range Sets (URS), particularly we study functions of the form fnf0, gng0 sharing a constant, counting or not multiplicities. Finally, we look for the existence of solutions of functional equations of Diophantine type: functional equation such as P(x) = Q(y) where P and Q are polynomials whose coe cients are meromorphic functions. Su cient conditions are given in order to show that there exist no pair of admissible solutions for such equations or in certain cases, solutions exist with a very particular form. The most used method is the p-adic Nevanlinna Theory which no only applies to ultrametric meromorphic functions in the whole eld K, but also applies to unbounded meromorphic functions inside an open disk. The most used theorems are: the Ultrametric Nevanlinna's second main theorem, the Ultrametric Nevanlinna's theorem on 3 small functions and the Ultrametric Milloux's inequality.
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Jacqueline Alejandra Ojeda Fuentealba. Distribution de valeurs des fonctions méromorphes ultramétriques, application de la théorie de Nevanlinna. Mathématiques générales [math.GM]. Université Blaise Pascal - Clermont-Ferrand II, 2008. Français. ⟨tel-00730984⟩

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