Reduction of Elementary Integrability of Polynomial Vector Fields

Abstract

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.