Reduction of Elementary Integrability of Polynomial Vector Fields

TL;DR

This paper extends classical results on elementary integrability of polynomial vector fields, showing that such systems possess multiple algebraic first integrals and rational Jacobian multipliers, broadening understanding of their integrability properties.

Contribution

It generalizes Prelle and Singer's results to higher dimensions, demonstrating the existence of multiple algebraic first integrals and rational Jacobian multipliers for elementary integrable polynomial vector fields.

Findings

Any n-dimensional elementary integrable polynomial vector field has n-1 algebraic first integrals.

Such vector fields have a rational Jacobian multiplier.

Extension of classical 2D results to higher dimensions.

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.