Two continuous maps $f, g : \mathbb{ C}^2\to\mathbb{ C}^2$ are said to be topologically equivalent if there exist homeomorphisms $\varphi,\psi:\mathbb{ C}^2\to\mathbb{ C}^2$ satisfying $\psi\circ f\circ\varphi = g$. 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 $d=4$.
Categories: math.AG math.CO