Difference Picard theorem for meromorphic functions of several variables

TL;DR

This paper extends Picard's theorem to meromorphic functions of several variables with low hyper-order, showing conditions under which such functions are periodic, and develops difference analogues of key Nevanlinna theory results.

Contribution

It generalizes Picard's theorem for meromorphic functions of several variables with hyper-order less than 2/3 and introduces difference analogues of classical Nevanlinna theory results.

Findings

Meromorphic functions with certain invariant pre-images are periodic.

Established difference versions of the lemma on the logarithmic derivative.

Derived difference analogues of the second main theorem.

Abstract

It is shown that if three distinct values of a meromorphic function f:C^n -> P^1 of hyper-order strictly less than 2/3 have forward invariant pre-images with respect to a translation t:C^n -> C^n, t(z)=z+c, then f is a periodic function with period c. This result can be seen as a generalization of M. Green's Picard-type theorem in the special case where the hyper-order of f is less than 2/3, since the empty pre-images of the usual Picard exceptional values are by definition always forward invariant. In addition, difference analogues of the lemma on the logarithmic derivative and of the second main theorem of Nevanlinna theory for meromorphic functions C^n -> P^1 are given, and their applications to partial difference equations are discussed.