### 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 $F$.

$$

\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 $F$.

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Journal of Mathematics Research ISSN 1916-9795 (Print) ISSN 1916-9809 (Online)

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