On the density of image of differential operators generated by polynomials

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The main result of this work is the following theorem: LetE denote the ring of entire functions on C2. LetP∈C[x, y]. Forf∈E, set 1 $$D_P (f) = \frac{{\partial P}}{{\partial x}}\frac{{\partial f}}{{\partial y}} - \frac{{\partial P}}{{\partial y}}\frac{{\partial f}}{{\partial x}}$$ .Theorem.The image of D p is dense in E if and only ifP=σ(x), where σ is an automorphism of C[x, y].