William Cherry

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.

UNIQUENESS POLYNOMIALS FOR ENTIRE FUNCTIONS

A polynomial P is called a strong uniqueness polynomial for the family of non-constant entire functions on the complex plane if one cannot find two distinct non-constant entire functions f and g and a non-zero constant c such that P(f)=cP(g). We give necessary and sufficient geometric conditions on the divisor of zeros of P that P be a strong uniqueness polynomial for the family of non-constant entire functions on the complex plane.

Generalized ABC theorems for non-Archimedean entire functions of several variables in arbitrary characteristic

We prove generalized ABC theorems for vanishing sums of non-Archimedean entire functions of several variables in arbitrary characteristic.

The electron mechanics of induction acceleration

Abstract The radial and axial motions of electrons in the betatron are described by means of a potential function of forces. Previously reported conditions of equilibrium, stability and damping of oscillations are derived for the region of parabolic variation of the potential. Extension of the analysis to, non-parabolic regions gives an account of the injection in conventional instruments in better agreement with experiment, particularly in regard to higher voltages of injection.Space charge limitations are discussed with the help of the Laplacian of the potential of forces.By means of an additional radial electric field electrons can be introduced as in the magnetron, without any asymmetry inherent in the conventional betatron circumferential injector. The analysis of the conditions of equilibrium and stability, greatly facilitated in this case by the notion of potential, shows that no substantial improvement in space charge limitations can be expected and that the required variations between the flux linking the electron orbits and the magnetic and electric fields at the orbits are difficult to realize on account of their complexity and narrow tolerances. The X-ray output of a small experimental double yoke instrument was measured by a phototube multiplier viewing an irradiated fluorescent screen and gave evidence of multiple group electron capture.

Algebraic degeneracy of non-archimedean analytic maps

Abstract We prove non-Archimedean analogs of results of Noguchi and Winkelmann showing algebraic degeneracy of rigid analytic maps to projective varieties omitting an effective divisor with sufficiently many irreducible components relative to the rank of the group they generate in the Neron-Severi group of the variety.

Some projective distance inequalities for simplices in complex projective space

We prove inequalities relating the absolute value of the determinant of n+1 linearly independent unit vectors in an n+1 dimensional complex vector space and the projective distances from the vertices to the hyperplanes containing the opposite faces of the simplices in complex projective n-space whose vertices or faces are determined by the given vectors.

Supplement and Erratum to “Algebraic degeneracy of non-Archimedean analytic maps” [Indag. Math. (N.S.) 19 (2008) 481–492]

We prove non-Archimedean analogs of results of Noguchi and Winkelmann showing algebraic degeneracy of rigid analytic maps to projective varieties omitting an effective divisor with sufficiently many irreducible components relative to the rank of the group they generate in the Néron-Severi group of the variety.

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.