Meromorphic Functions over Non-Archimedean Fields

Häftad, Engelska, 2010

AvPei-Chu Hu,Chung-Chun Yang

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This book introduces value distribution theory over non-Archimedean fields, starting with a survey of two Nevanlinna-type main theorems and defect relations for meromorphic functions and holomorphic curves. Secondly, it gives applications of the above theory to, e.g., abc-conjecture, Waring's problem, uniqueness theorems for meromorphic functions, and Malmquist-type theorems for differential equations over non-Archimedean fields. Next, iteration theory of rational and entire functions over non-Archimedean fields and Schmidt's subspace theorems are studied. Finally, the book suggests some new problems for further research. Audience: This work will be of interest to graduate students working in complex or diophantine approximation as well as to researchers involved in the fields of analysis, complex function theory of one or several variables, and analytic spaces.

Produktinformation

Utgivningsdatum2010-12-07
Mått155 x 235 x 17 mm
Vikt464 g
FormatHäftad
SpråkEngelska
SerieMathematics and Its Applications
Antal sidor295
FörlagSpringer
ISBN9789048155460

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1 Basic facts in p-adic analysis.- 1.1 p-adic numbers.- 1.2 Field extensions.- 1.3 Maximum term of power series.- 1.4 Weierstrass preparation theorem.- 1.5 Newton polygons.- 1.6 Non-Archimedean meromorphic functions.- 2 Nevanlinna theory.- 2.1 Characteristic functions.- 2.2 Growth estimates of meromorphic functions.- 2.3 Two main theorems.- 2.4 Notes on the second main theorem.- 2.5 ‘abc’ conjecture over function fields.- 2.6 Waring’s problem over function fields.- 2.7 Exponent of convergence of zeros.- 2.8 Value distribution of differential polynomials.- 3 Uniqueness of meromorphic functions.- 3.1 Adams-Straus’ uniqueness theorems.- 3.2 Multiple values of meromorphic functions.- 3.3 Uniqueness polynomials of meromorphic functions.- 3.4 Unique range sets of meromorphic functions.- 3.5 The Frank-Reinders’ technique.- 3.6 Some urscm for M(?) and A(?).- 3.7 Some ursim for meromorphic functions.- 3.8 Unique range sets for multiple values.- 4 Differential equations.- 4.1 Malmquist-type theorems.- 4.2 Generalized Malmquist-type theorems.- 4.3 Further results on Malmquist-type theorems.- 4.4 Admissible solutions of some differential equations.- 4.5 Differential equations of constant coefficients.- 5 Dynamics.- 5.1 Attractors and repellers.- 5.2 Riemann-Hurwitz relation.- 5.3 Fixed points of entire functions.- 5.4 Normal families.- 5.5 Montel’s theorems.- 5.6 Fatou-Julia theory.- 5.7 Properties of the Julia set.- 5.8 Iteration of z ? zd.- 5.9 Iteration of z ? z2 + c.- 6 Holomorphic curves.- 6.1 Multilinear algebra.- 6.2 The first main theorem of holomorphic curves.- 6.3 The second main theorem of holomorphic curves.- 6.4 Nochka weight.- 6.5 Degenerate holomorphic curves.- 6.6 Uniqueness of holomorphic curves.- 6.7 Second main theorem for hypersurfaces.- 6.8Holomorphic curves into projective varieties.- 7 Diophantine approximations.- 7.1 Schmidt’s subspace theorems.- 7.2 Vojta’s conjecture.- 7.3 General subspace theorems.- 7.4 Ru-Vojta’s subspace theorem for moving targets.- 7.5 Subspace theorem for degenerate mappings.- A The Cartan conjecture for moving targets.- A.1 Non-degenerate holomorphic curves.- A.2 The Steinmetz lemma.- A.3 A defect relation for moving targets.- A.4 The Ru-Stoll techniques.- A.5 Growth of the Steinmetz-Stoll mappings.- A.6 Moving targets in subgeneral position.- A.7 Moving targets in general position.- Symbols.