2025/12/17 by Aziz, Waleed, Christopher, Colin, Pantazi, Chara +1
Mathematics · Computer Science · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Polynomial and algebraic computation #Nonlinear Waves and Solitons
paper · doi:10.48550/arxiv.2512.15522
We consider complex rational vector fields in dimension n>2 (equivalently, differential forms of degree n-1 in n variables) which admit a Liouvillian first integral. Extending a classical result by Singer for n=2, our main result states that there exists a first integral which is obtained by two successive integrations from one-forms with coefficients in a finite algebraic extension of the rational function field. The proof uses Puiseux series in a novel way to simplify computations. We also apply this method to give elementary proofs of Singer's theorem for rational one-forms, and of the Prelle-Singer theorem on elementary integrability of rational vector fields.