Integrating the Jacobian equation

TL;DR

This paper demonstrates that a specific differential equation involving polynomials in two variables can be algebraically integrated by expressing the homogeneous components of the involved polynomials, simplifying the solution process.

Contribution

It provides a method to algebraically integrate the Jacobian equation for polynomial pairs, explicitly relating the homogeneous components of the solutions.

Findings

The differential equation can be transformed into an algebraic system.

Homogeneous components of Q are explicitly expressed in terms of P.

Remaining equations involve only P's homogeneous components.

Abstract

We show essentially that the differential equation $\frac{\partial (P,Q)}{\partial (x,y)} =c \in {\mathbb C}$, for $P,\,Q \in {\mathbb C}[x,y]$, may be "integrated", in the sense that it is equivalent to an algebraic system of equations involving the homogeneous components of $P$ and $Q$. Furthermore, the first equations in this system give explicitly the homogeneous components of $Q$ in terms of those of $P$. The remaining equations involve only the homogeneous components of $P$.