VALUE DISTRIBUTION IN P-ADIC ANALYSIS

by Alain Escassut

Overview

The book first explains the main properties of analytic functions in order to use them in the study of various problems in p-adic value distribution. Certain properties of p-adic transcendental numbers are examined such as order and type of transcendence, with problems on p-adic exponentials. Lazard's problem for analytic functions inside a disk is explained. P-adic meromorphics are studied. Sets of range uniqueness in a p-adic field are examined. The ultrametric Corona problem is studied. Injective analytic elements are characterized. The p-adic Nevanlinna theory is described and many applications are given: p-adic Hayman conjecture, Picard's values for derivatives, small functions, branched values, growth of entire functions, problems of uniqueness, URSCM and URSIM, functions of uniqueness, sharing value problems, Nevanlinna theory in characteristic p>0, p-adic Yosida's equation.

Product Details

ISBN-13:	9789814730129
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	11/27/2015
Sold by:	Barnes & Noble
Format:	eBook
Pages:	560
File size:	48 MB
Note:	This product may take a few minutes to download.

Table of Contents

Chapter 1 Ultrametric fields 1

Chapter 2 Monotonous and circular filters 10

Chapter 3 Ultrametric absolute values for rational functions 20

Chapter 4 Hensel Lemma 29

Chapter 5 Extensions of ultrametric fields: the field C_P 34

Chapter 6 Normal extensions of Q_p inside C_P 41

Chapter 7 Spherically complete extensions 48

Chapter 8 Transcendence order over Q_P in C_P 52

Chapter 9 Transcendence type over Q in C_P 59

Chapter 10 Algebras R(D) 66

Chapter 11 Analytic elements 72

Chapter 12 Composition of analytic elements 79

Chapter 13 Mult(H(D), U_D) 84

Chapter 14 Power and Laurent series 88

Chapter 15 Mittag-Leffler Theorem and dual of a space H(D) 97

Chapter 16 Factorization of analytic elements 108

Chapter 17 Algebras H(D) 112

Chapter 18 Derivative of analytic elements 119

Chapter 19 Properties of the function Ψ for analytic elements 131

Chapter 20 Vanishing along a monotonous filter 136

Chapter 21 Quasi-minorated elements 142

Chapter 22 Zeros of power series 148

Chapter 23 Maximum principle 159

Chapter 24 Image of a disk 163

Chapter 25 Logarithm and exponential in a p-adic field 174

Chapter 26 Problems on p-adic exponentials 178

Chapter 27 Quasi-invertible analytic elements 183

Chapter 28 Divisors of analytic functions 191

Chapter 29 Michel Lazard's problem 201

Chapter 30 Sets of range uniqueness in p-adic fields 208

Chapter 31 Motzkin factorization, roots of analytic functions 220

Chapter 32 Meromorphic functions 237

Chapter 33 Residues of meromorphic functions 245

Chapter 34 Identity sequences for p-adic functions 253

Chapter 35 The set Mult(A_b(d(0,R^-)), . _d(0,R^-)) 259

Chapter 36 The Corona problem on A_b(d(0,1^-)) 266

Chapter 37 Shilov boundary for algebras H(D) 276

Chapter 38 Mappings from Φ(D) to the tree Φ(K) 288

Chapter 39 Injective analytic elements 296

Chapter 40 Nevanlinna Theory 308

Chapter 41 Immediate applications of Nevanlinna Theory 319

Chapter 42 Applications to curves 324

Chapter 43 Small functions 332

Chapter 44 Exceptional values of functions and derivatives 341

Chapter 45 The p-adic Hayman conjecture 351

Chapter 46 Optimal functions 362

Chapter 47 Order of growth for entire functions 367

Chapter 48 Type of growth for entire functions 375

Chapter 49 Growth of the derivative of an entire function 386

Chapter 50 Branched values 391

Chapter 51 Affinely rigid sets 404

Chapter 52 Composition of meromorphic functions 410

Chapter 53 Functions of uniqueness 417

Chapter 54 Urscm and ursim 429

Chapter 55 Other urscm, ursim and non-urscm 442

Chapter 56 Functions f'P'(f) 451

Chapter 57 Functions sharing values 462

Chapter 58 Analytic functions sharing a small function 469

Chapter 59 Meromorphic functions sharing a small function 473

Chapter 60 Nevanlinna Theory in characteristic p 488

Chapter 61 Strong uniqueness and URSCM in characteristic p 495

Chapter 62 The Functional Equation P(f) = Q(g) 500

Chapter 63 Rational decomposition for entire functions 508

Chapter 64 Yosida's equation 511

Chapter 65 Yoshida's equation inside a disk 515

Bibliography 523

Definitions 533

Notations 539