Algebraic curves and meromorphic functions Sharing pairs of values

TL;DR

This paper characterizes solutions to a complex analysis problem involving meromorphic functions sharing value pairs, showing they correspond to low-genus algebraic curves, thus enabling computational approaches.

Contribution

It provides a novel algebraic characterization of meromorphic function pairs sharing values, linking the problem to algebraic curves and facilitating computational solutions.

Findings

Solutions parametrize genus zero algebraic curves

The problem reduces to low-degree algebraic curves with five parameters

Enables computer algebra methods to analyze the problem

Abstract

The 4IM+1CM problem is determining all pairs (f,g) of meromorphic functions in the complex plane that are not Moebius transformations of each other and share five pairs of complex values, one of them counting multiplicities. It is shown that every solution to this problem parametrizes an algebraic curve of genus zero and low degree depending on five parameters only, which enables handing over the problem to computer algebra specialists.