Generalized ODE reduction algorithm with bounded degree transformation

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}.