Computational study demonstrates an efficient reduction algorithm for two-dimensional polynomial vector fields, highlighting practical simplification to generalized Abel equations.
As a generalization of our previous result, this paper aims to answer the following question: given a 2-dimensional polynomial vector field y'=M(x,y)/N(x,y), how to find a rational transformation y -> A(x,y)/B(x,y) with bounded degree numerator, the inverse of which transforms this vector field into a simpler form y' = sumᵢ₌₀^n fi(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 on ResearchGate
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A 2026 study studied this question.
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