VALUE DISTRIBUTION IN ULTRAMETRIC ANALYSIS AND APPLICATIONS

After a construction of the complete ultrametric fields K, the book presents most of properties of analytic and meromorphic functions in K: algebras of analytic elements, power series in a disk, order, type and cotype of growth of entire functions, clean functions, question on a relation true for clean functions, and a counter-example on a non-clean function. Transcendence order and transcendence type are examined with specific properties of certain p-adic numbers. The Kakutani problem for the 'corona problem' is recalled and multiplicative semi-norms are described. Problems on exponential polynomials, meromorphic functions are introduced and the Nevanlinna Theory is explained with its applications, particularly to problems of uniqueness. Injective analytic elements and meromorphic functions are examined and characterized through a relation. The Nevanlinna Theory out of a hole is described. Many results on zeros of a meromorphic function and its derivative are examined, particularly the solution of the Hayman conjecture in a P-adic field is given. Moreover, if a meromorphic functions in all the field, admitting primitives, admit a Picard value, then it must have enormously many poles. Branched values are examined, with links to growth order of the denominator. The Nevanlinna theory on small functions is explained with applications to uniqueness for a pair of meromorphic functions sharing a few small functions. A short presentation in characteristic p is given with applications on Yoshida equation. Contents: Affinely Rigid Sets Properties of Ultrametric Fields Monotonous and Circular Filters Ultrametric Absolute Values for Rational Functions Hensel Lemma Extensions of Ultrametric Fields: The Field ℂp Normal Extensions of ℚp Inside ℂp Spherically Complete Extensions Transcendence Order and Transcendence Type Algebras R(D) Analytic Elements Composition of Analytic Elements Multiplicative Spectrum of H(D) Power and Laurent Series Krasner Mittag-Leffler Theorem Factorization of Analytic Elements Algebras H(D) Derivative of Analytic Elements Properties of the Function Ψ for Analytic Elements Vanishing Along a Monotonous Filter Quasi-Minorated Elements Zeros of Power Series Image of a Disk Quasi-Invertible Analytic Elements Logarithm and Exponential in a p-adic Field Problems on p-adic Exponentials Divisors of Analytic Functions Michel Lazard's Problem Motzkin Factorization and Roots of Analytic Functions Order of Growth for Entire Functions Type of Growth for Entire Functions Growth of the Derivative of an Entire Function Growth of an Analytic Function in an Open Disk The Set Mult(Ab(d(0,R–)) The Corona Problem on Ab(d(0, 1–)) Meromorphic Functions in 𝕂 Residues of Meromorphic Functions Bezout Algebras of Analytic Functions Meromorphic Functions Out of a Hole Shilov Boundary for Algebras H(D) Mappings from Φ(D) to the Tree Φ(𝕂) Injective Analytic Elements Counting Functions and Nevanlinna Theory A Non-Clean Entire Function Nevanlinna Theory in 𝕂 and Inside a Disk Nevanlinna Theory Out of a Hole Immediate Applications of the Nevanlinna Theory Applications to Curves Branched Values Exceptional Values of Functions and Derivatives Small Functions Meromorphic Functions Sharing Some Small Ones The p-adic Hayman Conjecture Composition of Meromorphic Functions Functions of Uniqueness Urscm and Ursim Other Urscm, Usim, and Non-Urscm Nevanlinna Theory in Characteristic p Strong Uniqueness and URSCM in Characteristic p The Functional Equation P(ƒ) = Q(g) Yoshida's Equation in the Field 𝕂 Yoshida's Equation Inside a Disk Readership: Undergraduate researchers in ultrametric analysis and all researchers in ultrametric analysis. The book may be used in a course of Master or preparation of a. Doctorate. Researchers in number theory, researchers in physics using p-adic numbers. Alain Escassut obtained a Doctorate in 1970 and a Doctorat d'Etat in 1972 at Université Bordeaux 1 and worked there from 1969 to 1987 (he was Visiting-Assistant...