# Using discrete Darboux polynomials to detect and determine preserved measures and integrals of rational maps

## TL;DR

This paper introduces a systematic symbolic algebra-based method using Discrete Darboux Polynomials to detect and compute preserved measures and integrals of rational maps, enabling comprehensive analysis with sufficient computational resources.

## Contribution

The paper presents a novel approach leveraging cofactors and Discrete Darboux Polynomials for identifying conserved quantities in rational maps, advancing the computational detection process.

## Key findings

- Successfully detects preserved measures and integrals in example rational maps.
- Demonstrates the method's effectiveness with symbolic algebra tools.
- All rational preserved integrals can be found given enough computational power.

## Abstract

In this Letter we propose a systematic approach for detecting and calculating preserved measures and integrals of a rational map. The approach is based on the use of cofactors and Discrete Darboux Polynomials and relies on the use of symbolic algebra tools. Given sufficient computing power, all rational preserved integrals can be found. We show, in two examples, how to use this method to detect and determine preserved measures and integrals of the considered rational maps.