P-Adic Analytic Functions

图书简介

P-adic Analytic Functions describes the definition and properties of p-adic analytic and meromorphic functions in a complete algebraically closed ultrametric field. Various properties of p-adic exponential-polynomials are examined, such as the Hermite Lindemann theorem in a p-adic field, with a new proof. The order and type of growth for analytic functions are studied, in the whole field and inside an open disk. P-adic meromorphic functions are studied, not only on the whole field but also in an open disk and on the complemental of a closed disk, using Motzkin meromorphic products. Finally, the p-adic Nevanlinna theory is widely explained, with various applications. Small functions are introduced with results of uniqueness for meromorphic functions. The question of whether the ring of analytic functions — in the whole field or inside an open disk — is a Bezout ring is also examined.

Ultrametric Fields: Basic Definitions and 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; Analytic Elements and Analytic Functions: Algebras R(D); Analytic Elements; Composition of Analytic Elements; Multiplicative Spectrum of H(D); Power and Laurent Series; Krasner–Mittag–Leffer 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 Functions in an Open Disk; Meromorphic Functions and Nevanlinna Theory: Meromorphic Functions in ?? Residues of Meromorphic Functions; Meromorphic Functions Out of a Hole; Nevanlinna Theory in ?? and in an Open Disk; Nevanlinna Theory Out of a Hole; Immediate Applications of the Nevanlinna Theory; Branched Values; Exceptional Values of Functions and Derivatives; The p-adic Hayman Conjecture; Small Functions; Bibliography; Definitions; Notations;

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P-Adic Analytic Functions P-ADIC解析函数 数论

P-Adic Analytic Functions

P-ADIC解析函数
数论