On the density of image of differential operators generated by polynomials (Q1823998)

Let P(x,y) be a polynomial in two variables (over the complex numbers, say) and define the partial differential operator \[ D_ P(x,y)=\frac{\partial P}{\partial x}\frac{\partial f}{\partial y}- \frac{\partial P}{\partial y}\frac{\partial f}{\partial x}. \] The author proves that the image of \(D_ P\) is dense in the ring of entire functions on \({\mathbb{C}}^ 2\) if and only if P is a generator for the ring \({\mathbb{C}}[x,y]\). - Obviously the result is related to the Jacobian conjecture, and also has intrinsic interested. The proof proceeds by a careful analysis of critical points of a polynomial of two variables.