Generalized ODE reduction algorithm with bounded degree transformation

TL;DR

This paper presents a generalized algorithm for transforming 2D polynomial vector fields into simpler forms using bounded degree rational transformations, enhancing understanding of their properties.

Contribution

It introduces a new algorithm for finding rational transformations with bounded degree to simplify polynomial vector fields, extending previous work.

Findings

Algorithm effectively simplifies complex vector fields

Implementation demonstrates good performance

Code is publicly available for use and further research

Abstract

As a generalization of our previous result\cite{huang2025algorithm}, this paper aims to answer the following question: Given a 2-dimensional polynomial vector field $y^{\prime}=\frac{M(x,y)}{N(x,y)}$, how to find a rational transformation $y \to \frac{A(x,y)}{B(x,y)}$ with bounded degree numerator, the inverse of which transforms this vector field into a simpler form $y^{\prime}=\sum_{i=0}^nf_i(x)y^i$. Such a structure, often known as the generalized Abel equation and has been studied in various areas, provides a deeper insight into the property of the original vector field. We have implemented an algorithm with considerable performance to tackle this problem and the code is available in \href{https://www.researchgate.net/publication/393362858_Generalized_ODE_reduction_algorithm}{Researchgate}.