# Rationality is decidable for nearly Euclidean Thurston maps

## TL;DR

This paper proves that the problem of deciding whether a nearly Euclidean Thurston map is equivalent to a rational function is decidable, providing bounds that explain the effectiveness of computational tools like NETmap.

## Contribution

It establishes bounds on the size of the finite computation needed to determine rationality of NET maps, linking geometric size to algorithmic decidability.

## Key findings

- Decidability of rationality for NET maps is established.
- Bounds are provided for the computational complexity based on diagram size.
- The results explain the practical success of the NETmap software.

## Abstract

Nearly Euclidean Thurston (NET) maps are described by simple diagrams which admit a natural notion of size. Given a size bound $C$, there are finitely many diagrams of size at most $C$. Given a NET map $F$ presented by a diagram of size at most $C$, the problem of determining whether $F$ is equivalent to a rational function is, in theory, a finite computation. We give bounds for the size of this computation in terms of $C$ and one other natural geometric quantity. This result partially explains the observed effectiveness of the computer program NETmap in deciding rationality.