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.