Book•
Nevanlinna Theory and Its Relation to Diophantine Approximation
01 Jun 2001-
TL;DR: Theorem of Faltings complex Hyperbolic Manifolds and Lang's Conjecture as mentioned in this paper is related to the moving target problems in the context of meromorphic functions.
Abstract: Nevanlinna Theory for Meromorphic Functions and Roth's Theorem Holomorphic Curves into Compact Riemann Surfaces and Theorems of Siegel, Roth, and Faltings Holomorphic Curves in Pn(C) and Schmidt's Sub-Space Theorem The Moving Target Problems Equi-Dimensional Nevanlinna Theory and Vojta's Conjecture Holomorphic Curves in Abelian Varieties and the Theorem of Faltings Complex Hyperbolic Manifolds and Lang's Conjecture.
Citations
More filters
Book•
13 Aug 2008TL;DR: A survey of results after 1970 can be found in this paper, where the authors present a survey of meromorphic functions of finite-order functions with respect to Riemann surfaces.
Abstract: Characteristics of the behavior of a meromorphic function and the first fundamental theorem Meromorphic functions of finite order The second fundamental theorem Deficient values Asymptotic properties of meromorphic functions and deficiencies Value distribution with respect to the arguments Applications of Riemann surfaces to value distribution On the magnitude of an entire function Notes A survey of some results after 1970 Bibliography References added to the English edition Author index Subject index Notation index.
242 citations
TL;DR: In this paper, it is argued that the integrability of many difference equations is related to the structure of their solutions at infinity in the complex plane and that Nevanlinna theory provides many of the concepts necessary to detect integranability in a large class of equations.
Abstract: It is well known that the integrability (solvability) of a differential equation is related to the singularity structure of its solutions in the complex domain - an observation that lies behind the Painleve test. A number of ways of extending this philosophy to discrete equations are explored. First, following the classical work of Julia, Birkhoff and others, a natural interpretation of these equations in the complex domain as difference or delay equations is described and it is noted that arbitrary periodic functions play an analogous role for difference equations to that played by arbitrary constants in the solution of differential equations. These periodic functions can produce spurious branching in solutions and are factored out of the analysis which concentrates on branching from other sources. Second, examples and theorems from the theory of difference equations are presented which show that, modulo these periodic functions, solutions of a large class of difference equations are meromorphic, regardless of their integrability. It is argued that the integrability of many difference equations is related to the structure of their solutions at infinity in the complex plane and that Nevanlinna theory provides many of the concepts necessary to detect integrability in a large class of equations. A perturbative method is then constructed and used to develop series in z and the derivative of log (z ), where z is the independent variable of the difference equation. This method provides an analogue of the series developed in the Painleve test for differential equations. Finally, the implications of these observations are discussed for two tests which have been studied in the literature regarding the integrability of discrete equations.
237 citations
TL;DR: In this article, a difference analogue of M. Green's Picard-type theorem for holomorphic curves is presented, which can be described as a difference analog of Green's first main theorem for the Casorati determinant and an extended version of the difference analogue on the logarithmic derivatives.
Abstract: If f : C ! P n is a holomorphic curve of hyper-order less than one for which 2n + 1 hyperplanes in general position have forward invariant preimages with respect to the translation �(z) = z +c, then f is periodic with period c 2 C. This result, which can be described as a difference analogue of M. Green's Picard-type theorem for holomorphic curves, follows from a more general result presented in this paper. The proof relies on a new version of Cartan's second main theorem for the Casorati determinant and an extended version of the difference analogue of the lemma on the logarithmic derivatives, both of which are proved here. Finally, an application to the uniqueness theory of meromorphic functions is given, and the sharpness of the obtained results is demonstrated by examples.
179 citations
01 Jan 2011
TL;DR: The potential theory in value distribution has been studied in the context of Meromorphic Functions with Radially Distributed Values (RDV) and Singular values as discussed by the authors, where the potential theory of value distribution is applied to the case of MRFs.
Abstract: Preliminaries of Real Functions.- Characteristics of a Meromorphic Function.- T Directions of a Meromorphic Function.- Argument Distribution and Deficient Values.- Meromorphic Functions With Radially Distributed Values.- Singular Values of Meromorphic Functions.- The Potential Theory in Value Distribution.
143 citations
Abstract: The Painlev´ e property is closely connected to differential equations that are integrable via related iso-monodromy problems. Many apparently integrable discrete analogues of the Painlev´ e equations have appeared in the literature. The existence of sufficiently many finite-order meromorphic solutions appears to be a good analogue of the Painlev´ e property for discrete equations, in which the independent variable is taken to be complex. A general introduction to Nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q-difference operators. New results are presented concerning equations of the form w(z +1 )w(z − 1) = R(z, w), where R is rational in w with meromorphic coefficients.
106 citations