| Introduction |
|
vii | |
|
|
|
1 | (52) |
|
A.1 Basic definitions and properties of ultrametric fields |
|
|
1 | (9) |
|
A.2 Monotonous and circular niters |
|
|
10 | (9) |
|
A.3 Ultrametric absolute values for rational functions |
|
|
19 | (9) |
|
|
|
28 | (5) |
|
A.5 Extensions of ultrametric fields: The field Cp |
|
|
33 | (6) |
|
A.6 Normal extensions of Qp inside Cp |
|
|
39 | (4) |
|
A.7 Spherically complete extensions |
|
|
43 | (4) |
|
A.8 Transcendence order and transcendence type |
|
|
47 | (6) |
|
B Analytic elements and analytic functions |
|
|
53 | (180) |
|
|
|
53 | (5) |
|
|
|
58 | (6) |
|
B.3 Composition of analytic elements |
|
|
64 | (5) |
|
B.4 Multiplicative spectrum of H(D) |
|
|
69 | (4) |
|
B.5 Power and Laurent series |
|
|
73 | (8) |
|
B.6 Krasner-Mittag-Leffler theorem |
|
|
81 | (10) |
|
B.7 Factorization of analytic elements |
|
|
91 | (4) |
|
|
|
95 | (6) |
|
B.9 Derivative of analytic elements |
|
|
101 | (12) |
|
B.10 Properties of the function * for analytic elements |
|
|
113 | (4) |
|
B.11 Vanishing along a monotonous filter |
|
|
117 | (5) |
|
B.12 Quasi-minorated elements |
|
|
122 | (6) |
|
B.13 Zeros of power series |
|
|
128 | (10) |
|
|
|
138 | (11) |
|
B.15 Quasi-invertible analytic elements |
|
|
149 | (7) |
|
B.16 Logarithm and exponential in a p-adic field |
|
|
156 | (4) |
|
B.17 Problems on p-adic exponentials |
|
|
160 | (8) |
|
B.18 Divisors of analytic functions |
|
|
168 | (10) |
|
B.19 Michel Lazard's problem |
|
|
178 | (7) |
|
B.20 Motzkin factorization and roots of analytic functions |
|
|
185 | (16) |
|
B.21 Order of growth for entire functions |
|
|
201 | (9) |
|
B.22 Type of growth for entire functions |
|
|
210 | (9) |
|
B.23 Growth of the derivative of an entire function |
|
|
219 | (4) |
|
B.24 Growth of an analytic function in an open disk |
|
|
223 | (10) |
|
C Meromorphic functions and Nevanlinna theory |
|
|
233 | (86) |
|
C.1 Meromorphic functions in K |
|
|
233 | (7) |
|
C.2 Residues of meromorphic functions |
|
|
240 | (7) |
|
C.3 Meromorphic functions out of a hole |
|
|
247 | (2) |
|
C.4 Nevanlinna theory in K and in an open disk |
|
|
249 | (11) |
|
C.5 Nevanlinna theory out of a hole |
|
|
260 | (8) |
|
C.6 Immediate applications of the Nevanlinna theory |
|
|
268 | (5) |
|
|
|
273 | (8) |
|
C.8 Exceptional values of functions and derivatives |
|
|
281 | (8) |
|
|
|
289 | (14) |
|
C.10 The p-adic Hayman conjecture |
|
|
303 | (10) |
|
C.11 Bezout algebras of analytic functions |
|
|
313 | (6) |
| Bibliography |
|
319 | (6) |
| Definitions |
|
325 | (6) |
| Notations |
|
331 | (6) |
| Index |
|
337 | |
Lisainfo e-raamatute kohta