2024/02/14 by Hilany, Boulos El, Rose, Kemal
#58K15 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Primary 14D06 #Secondary: 14T20
paper · doi:10.48550/arxiv.2402.08993
Two continuous maps f, g : ℂ2→ℂ2 are said to be topologically equivalent if there exist homeomorphisms φ,ψ:ℂ2→ℂ2 satisfying ψ∘ f∘φ= g. It is known that there are finitely many topologically non-equivalent polynomial maps ℂ2→ℂ2 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.