Nevanlinna Theory in characteristic p and applications
by
Alain Escassut
Universite Blaise Pascal, Clermont-Ferrand, France
Coauthors: Abdelbaki Boutabaa

Let K be a complete ultrametric algebraically closed field of characteristic p. We show that Nevanlinna's Theorems hold, particularly the second main Theorem, without however some corrections. Many results obtained in characteristic zero have generalization. Many algebraic curves admit no parametrizations by meromorphic functions in K, or by unbounded meromorphic functions inside a disk, like in zero characteristic, provided we assume one of the function to have a non zero derivative. In functional equations f^m+gⁿ=1, conclusions also are similar to those obtained in zero characteristic, provided we replace m, n by [m\tilde]=m|m|_p,  [n\tilde]=n|n|_p. About Yoshida's equation with constant coefficients, all solutions are characterized: they generalize those obtained in zero characteristic, but with a more general form involving polynomials with a zero derivative. Finally we consider the (abc)-problem for p-adic entire functions (previously studied by P.C. Hu and C.-C. Yang in zero characteristic) and show that conclusions can be generalized (with a slight improvement) to fields of any characteristic p, with a correction depending on ramification indices. In a common work with William Cherry, we also study unique range sets for entire functions, counting multiplicities, in the field K, and show that, when (n, p)=1, affinely rigid sets of n points are the urs of n points. Besides, for all n >= 6 there exist urs of n points whenever p. Counter-examples are constructed. If p=3, there exist no urs of 3 points. All results have application to polynomials in fields of characteristic p.

Date received: July 20, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahv-78.