Some achievements of Nevanlinna theory

Some Results in View of Nevanlinna Theory

Lectures on Nevanlinna theory

Value distribution of a rational function f is controlled by its degree d, which is the number of preimages of a generic point. If we denote by n(a) the number of solutions of the equation f(z) = a, counting multiplicity, in the complex plane C, then n(a) ≤ d for all a ∈ C with equality for all a with one exception, namely a = f(∞). The number of critical points of f in C, counting multiplicity...

Nevanlinna Theory and Rational Points

S. Lang [L] conjectured in 1974 that a hyperbolic algebraic variety defined over a number field has only finitely many rational points, and its analogue over function fields. We discuss the Nevanlinna-Cartan theory over function fields of arbitrary dimension and apply it for Diophantine property of hyperbolic projective hypersurfaces (homogeneous Diophantine equations) constructed by Masuda-Nog...

Parametric Nevanlinna-Pick Interpolation Theory

We consider the robust control problem for the system with real uncertainty. This type of problem can be represented with some parameters varying between the boundaries and is formulated as parametric Nevanlinna-Pick interpolation problem in this paper. The existence of a solution for such interpolation problem depends on the positivity of the corresponding Pick matrix with elements belonging t...

Nevanlinna Theory and Diophantine Approximation

As observed originally by C. Osgood, certain statements in value distribution theory bear a strong resemblance to certain statements in diophantine approximation, and their corollaries for holomorphic curves likewise resemble statements for integral and rational points on algebraic varieties. For example, if X is a compact Riemann surface of genus > 1, then there are no non-constant holomorphic...