Integrating the Jacobian equation

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 $P$.