The polyhedral type of a polynomial map on the plane

Abstract

Two continuous maps $f,g:{\mathbb{C}}^2\to {\mathbb{C}}^2$ are said to be topologically equivalent if there exist homeomorphisms $\ddot{I}†,\ddot{I}ˆ:{\mathbb{C}}^2\to {\mathbb{C}}^2$ satisfying $\ddot{I}ˆ\circ f\circ \ddot{I}†=g$. It is known that there are finitely many topologically non-equivalent polynomial maps ${\mathbb{C}}^2\to {\mathbb{C}}^2$ with any given degree $d$. The number $T(d)$ of these topological types is known only whenever $d=2$. In this paper, we describe the topology of generic complex polynomial maps on the plane using the corresponding pair of Newton polytopes and establish a method for constructing topologically non-equivalent maps of degree $d$. We furthermore provide a software implementation of the resulting algorithm, and present lower bounds on $T(d)$ whenever $d=3$ and $d=4$.

30 pages, 6 figures, comments are welcome!