2021/11/21 by Thierry Combot, Combot, Thierry
Computer Science · Mathematics · #34A05 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2111.10809
openalex publication_date 2021/11/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a planar polynomial vector field X, and assume it admits a symbolic first integral F, i.e. of the 4 classes, in growing complexity: Rational, Darbouxian, Liouvillian and Riccati. If F is not rational, it is sometimes possible to reduce it to a simpler class first integral. We will present algorithms to reduce symbolic first integral to a lower complexity class. These algorithms allow to find the minimal class first integral and in particular to test the existence of a rational first integral except in the case where F is a k-Darbouxian first integral without singularities and k∈\2,3,4,6\. In this case, several examples are built and a procedure is presented which however requires the computation of elliptic factors in the Jacobian of a superelliptic curve.