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

TL;DR

This paper investigates Keller mappings in two complex variables, providing size estimates for their co-image and establishing a sufficient condition for surjectivity based on the Phantom curve, which relates to the asymptotic variety.

Contribution

It introduces a new approach to analyze the surjectivity of Keller mappings using the Phantom curve and estimates the co-image size, advancing understanding of polynomial mappings in complex variables.

Findings

Provides size estimates for the co-image of Keller mappings

Establishes a sufficient condition for surjectivity involving the Phantom curve

Links the Phantom curve to the asymptotic variety of the mapping

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 $F$.