Fermat functional equations over Riemann surfaces

TL;DR

This paper studies Fermat functional equations on open Riemann surfaces, establishing existence of solutions on hyperbolic surfaces and growth conditions on general surfaces based on the exponents.

Contribution

It provides new existence results for solutions on hyperbolic surfaces and growth constraints on solutions for general Riemann surfaces depending on exponents.

Findings

Existence of non-trivial solutions on hyperbolic Riemann surfaces.

Growth conditions for solutions on general Riemann surfaces.

Solutions depend on the size of the exponents.

Abstract

We investigate the existence of non-trivial holomorphic and meromorphic solutions of Fermat functional equations over an open Riemann surface $S$. When $S$ is hyperbolic, we prove that any $k$-term Fermat functional equation always exists non-trivial holomorphic and meromorphic solution. When $S$ is a general open Riemann surface, we prove that every non-trivial holomorphic or meromorphic solution satisfies a growth condition, provided that the power exponents of the equations are bigger than some certain positive integers.