# On the algebraicity of germs of meromorphic functions

## TL;DR

This paper proves that every germ of a meromorphic function at the origin in complex or real space can be transformed into an algebraic meromorphic function via biholomorphic change of variables, establishing a form of algebraic equivalence.

## Contribution

The paper demonstrates that all germs of analytic meromorphic functions in two variables are equivalent to algebraic ones under biholomorphic transformations, extending to real functions.

## Key findings

- Germs of meromorphic functions are algebraically equivalent after biholomorphic change.
- The result applies to both complex and real analytic meromorphic functions.
- This establishes a universal algebraic form for such germs.

## Abstract

In this article we prove that every germ of analytic meromorphic function at $(\mathbb{C}^{2},0)$ is equivalent, under the right composition by a germ of biholomorphism, to a germ of algebraic meromorphic function. An analogous result is also true for real analytic meromorphic functions.