P-adic Analytic Functions

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 an open 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.
Contents:

• Introduction

• 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 ??
  • Normal Extensions of ?? Inside ??
  • 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 ? 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
  • Small Functions
  • The p-Adic Hayman Conjecture
  • Bezout Algebras of Analytic Functions

• Bibliography

• Definitions

• Notations

• Index

Readership: The target audience consists of researchers and post-graduate students in ultrametric analysis and number theory. Also appropriate for researchers in Levi-Civita fields in p-adic physics.
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