# Rational Solutions of First-Order Algebraic Ordinary Difference Equations

## TL;DR

This paper introduces an algebraic geometric method to analyze rational solutions of first-order algebraic ordinary difference equations, providing degree bounds and an algorithm for their computation.

## Contribution

It presents a novel algebraic geometric approach, including degree bounds and a complete algorithm for finding rational solutions of autonomous first-order difference equations.

## Key findings

- Established an upper bound for degrees of rational solutions.
- Developed a complete algorithm for computing rational solutions.
- Provided a systematic approach for autonomous first-order difference equations.

## Abstract

We propose an algebraic geometric approach for studying rational solutions of first-order algebraic ordinary difference equations. For an autonomous first-order algebraic ordinary difference equations, we give an upper bound for the degrees of its rational solutions, and thus derive a complete algorithm for computing corresponding rational solutions.