2024/12/06 by Wenyong Huang, Xiang Zhang, Huang, Wenyong +1
Mathematics · #Advanced Differential Equations and Dynamical Systems
paper · pdf · doi:10.48550/arxiv.2412.04750
Prelle and Singer showed in 1983 that if a system of ordinary differential equations defined on a differential field K has a first integral in an elementrary field extension L of K, then it must have a first integral consisting of algebraic elements over K via their constant powers and logarithms. Based on this result they further proved that an elementary integrable planar polynomial differential system has an integrating factor which is a fractional power of a rational function. Here we extend their results and prove that any n dimensional elementary integrable polynomial vector field has n-1 functionally independent first integrals being composed of algebraic elements over K. Furthermore, using the Galois theory we prove that the vector field has a rational Jacobian multiplier.