The polyhedral type of a polynomial map on the plane

Two continuous maps f, g : C²² are said to be topologically equivalent if there exist homeomorphisms φ,ψ:C²² satisfying ψ∘ f∘φ = g. It is known that there are finitely many topologically non-equivalent polynomial maps C²² 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 $d=4$.