Zhuan Ye

On Nevanlinna's second main theorem in projective space

SummaryWe first prove a theorem concerning higher order logarithmic partial derivatives for meromorphic functions of several complex variables. Then we show the best nature of the second main theorem in Nevanlinna theory under two different assumptions of non-degeneracy of meromorphic mappingsf : ℂn → ℙn for arbitrary positive integersn andm. Moreover, we derive a upper bound of the error term in the second main theorem for meromorphic mappings of finite order. Finally, we demonstrate the sharpness of all upper bounds in our main theorems.

Non-Archimedean Nevanlinna theory in several variables and the non-Archimedean Nevanlinna inverse problem

Cartan’s method is used to prove a several variable, non-Archimedean, Nevanlinna Second Main Theorem for hyperplanes in projective space. The corresponding defect relation is derived, but unlike in the complex case, we show that there can only be finitely many non-zero non-Archimedean defects. We then address the non-Archimedean Nevanlinna inverse problem, by showing that given a set of defects satisfying our conditions and a corresponding set of hyperplanes in projective space, there exists a non-Archimedean analytic function with the given defects at the specified hyperplanes, and with no other defects. 1. History and Introduction Nevanlinna theory, broadly speaking, studies to what extent something like the Fundamental Theorem of Algebra holds for meromorphic functions. Unlike polynomials, transcendental meromorphic functions, in general, have infinitely many zeros. However, they have only finitely many zeros inside a disc of radius r. Therefore, in order to study the values of a meromorphic function, Nevanlinna theory associates to each meromorphic function f, three functions of r, the distance from the origin (for their precise definitions, see Nevanlinna [Ne 2]). The “characteristic” or “height” function Tf (r) measures the growth of f and should be thought of as the analogue of the degree of a polynomial. The “counting” function Nf (a, r) counts the number of times (as a logarithmic average) f takes on the value a in the disc of radius r. Finally, the “mean-proximity function” mf (a, r) measures how often, on average, f stays “close to” the value a on the circle of radius r. Nevanlinna proved two “main” theorems about these functions. The so-called “First Main Theorem” states that Tf(r) = mf (a, r) +Nf (a, r) +O(1), where the bounded term O(1) depends on f and a but not on r. This should be thought of as a substitute for the Fundamental Theorem of Algebra in the following sense. The First Main Theorem says that mf (a, r)+Nf (a, r) is essentially independent of the value a, and this is analogous to the fact that a polynomial takes on every finite value the same number of times counting multiplicity. The First Received by the editors October 14, 1995 and, in revised form, June 17, 1996. 1991 Mathematics Subject Classification. Primary 11J99, 11S80, 30D35, 32H30, 32P05.

Structural Instability of Exponential Functions

We first prove some equivalent statements on /-stability of families of critically finite entire functions. Then, with these in hand, a conjecture con- cerning stability of the family of exponential functions is affirmatively answered in some cases.

A class of second order differential equations

We study the differential equationf″=N(f)f′2+M(f)f′+L(f), whereL, M, N are rational functions, and prove that if the differential equation has a transcendental meromorphic solutionf with order,p(f)>2, then the differential equation must be one of nine forms; and, moreover, we construct examples showing the existence of these nine forms with a transcendental meromorphic solution.

An analogue of continued fractions in number theory for Nevanlinna theory

We show an analogue of continued fractions in approximation to irrational numbers by rationals for Nevanlinna theory. The analogue is a sequence of points in the complex plane which approaches a given finite set of points and at a given rate in the sense of Nevanlinna theory.

A Further Digression into Number Theory: Theorems of Roth and Khinchin

This chapter is our second digression into number theory. We saw in §1.8 that Jensen’s Formula (or the First Main Theorem) can be viewed as an analogue of the Artin-Whaples product formula in number theory. In the present chapter, we discuss a celebrated number theory theorem known as Roth’s Theorem, and we explain how this is analogous to a weak form of the Second Main Theorem. In fact, it was the formal similarity between Nevanlinna’s Second Main Theorem and Roth’s Theorem that led C. F. Osgood (see [Osg 1981] and [Osg 1985]) to the discovery of an analogy between Nevanlinna theory and Diophantine approximations. As we said previously, such an analogy was later, but independently, explored in greater depth by P. Vojta. In this chapter we will discuss, without proof, some of the key points in Vojta’s monograph [Vojt 1987], and we include Vojta’s so called “dictionary” relating Nevanlinna theory and number theory. This section is intended for the analytically inclined and is only intended to provide the most basic insight into the beautiful analogy between Nevanlinna theory and Diophantine approximation theory. By omitting proofs, we have tried to make this section less demanding on the reader than [Vojt 19871. However, a true appreciation for Vojta’s analogy cannot be obtained without also studying the proofs of the number theoretic analogues in their full generality. Any reader that is seriously interested in the connection between Nevanlinna theory and Diophantine approximation is highly encouraged to carefully read Vojta’s monograph [Vojt 1987].

The Second Main Theorem via Negative Curvature

Shortly after R. Nevanlinna’s first proof of the Second Main Theorem, Nevanlinna’s brother, F. Nevanlinna, gave a “geometric” proof of the Second Main Theorem. In this chapter, we give a geometric proof of the Second Main Theorem based on “negative curvature,” which broadly speaking, has the same overall structure as F. Nevanlinna’s proof, although in terms of details, the proof we present here bears much greater resemblance to the work of Ahlfors [Ahlf 1941]. In Chapter 4, we will give another proof of the Second Main Theorem that is closer to R. Nevanlinna’s original proof. Of course, neither the Nevanlinna brothers nor Ahlfors were interested in the exact structure of the error term. The error term we will present here is essentially due to P.-M. Wong [Wong 1989].

More on the Error Term

In this final chapter, we further explore the error term in the Second Main Theorem and the Logarithmic Derivative Lemma. In §7.1, we construct examples showing that in the general case, the error terms we have given in the previous chapters are essentially the best possible. In §7.2, we explain how by putting rather modest growth restrictions on the functions under consideration and by enlarging the exceptional set, the lower order terms in the error term can be further improved. Finally, in §7.3, we compute the precise form of the error term for some of the more familiar special functions.

The Second Main Theorem via Logarithmic Derivatives

In this chapter we give a logarithmic derivative based proof of the Second Main Theorem. R. Nevanlinna’s original proof of this theorem was along these lines, and the proof we give here is generally speaking similar to the proof given in Hayman’s book [Hay 1964]. Neither Nevanlinna nor Hayman were interested in the precise structure of the error term, and they did not use the refined logarithmic derivative estimates of Gol’dberg and Grinshtein, as we shall here. A sharp error term in the Second Main Theorem using this type of method was first obtained by Hinkkanen [Hink 1992]. The exact method of proof given here makes use of ideas from Ye’s refinement of Cartan’s method [Cart 1933], as in [Ye 1995].

The First Main Theorem

As we mentioned in the introduction, the basis for Nevanlinna’s theory are his two “main” theorems. This chapter discusses the first and easier of the two.