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＜電子ブック＞
Meromorphic Functions over Non-Archimedean Fields / by Pei-Chu Hu, Chung-Chun Yang
(Mathematics and Its Applications ; 522)

版	1st ed. 2000.
出版者	(Dordrecht : Springer Netherlands : Imprint: Springer)
出版年	2000
本文言語	英語
大きさ	VIII, 295 p. 1 illus : online resource
著者標目	*Pei-Chu Hu author Chung-Chun Yang author SpringerLink (Online service)
件　名	LCSH:Mathematical analysis LCSH:Functions of complex variables FREE:Analysis FREE:Functions of a Complex Variable FREE:Several Complex Variables and Analytic Spaces
一般注記	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 Nevanlinna theory (or value distribution theory) in complex analysis is so beautiful that one would naturally be interested in determining how such a theory would look in the non­ Archimedean analysis and Diophantine approximations. There are two "main theorems" and defect relations that occupy a central place in N evanlinna theory. They generate a lot of applications in studying uniqueness of meromorphic functions, global solutions of differential equations, dynamics, and so on. In this book, we will introduce non-Archimedean analogues of Nevanlinna theory and its applications. In value distribution theory, the main problem is that given a holomorphic curve f : C -+ M into a projective variety M of dimension n and a family 01 of hypersurfaces on M, under a proper condition of non-degeneracy on f, find the defect relation. If 01 n is a family of hyperplanes on M = r in general position and if the smallest dimension of linear subspaces containing the image f(C) is k, Cartan conjectured that the bound of defect relation is 2n - k + 1. Generally, if 01 is a family of admissible or normal crossings hypersurfaces, there are respectively Shiffman's conjecture and Griffiths-Lang's conjecture. Here we list the process of this problem: A. Complex analysis: (i) Constant targets: R. Nevanlinna[98] for n = k = 1; H. Cartan [20] for n = k > 1; E. I. Nochka [99], [100],[101] for n > k ~ 1; Shiffman's conjecture partially solved by Hu-Yang [71J; Griffiths-Lang's conjecture (open) HTTP:URL=https://doi.org/10.1007/978-94-015-9415-8

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