Picard theorems for Keller mappings in dimension two and the phantom curve

Abstract

Let $F=(P,Q)\in \mathbb{C}[X,Y{]}^2$ be a polynomial mapping over the complex field $\mathbb{C}$. Suppose that $$\det J_F(X,Y):=\frac{\partial P}{\partial X}\frac{\partial Q}{\partial Y}-\frac{\partial P}{\partial Y}\frac{\partial Q}{\partial X}=a\in {\mathbb{C}}^{\times }.$$ A mapping that satisfies the assumptions above is called a Keller mapping. In this paper we estimate the size of the co-image of $F$. We give a sufficient condition for surjectivity of Keller mappings in terms of its Phantom curve. This curve is closely related to the asymptotic variety of $F$.