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The polyhedral type of a polynomial map on the plane

Boulos El Hilany, Kemal Rose

Abstract

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

The polyhedral type of a polynomial map on the plane

Abstract

Two continuous maps are said to be topologically equivalent if there exist homeomorphisms satisfying . It is known that there are finitely many topologically non-equivalent polynomial maps with any given degree . The number of these topological types is known only whenever . 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 . We furthermore provide a software implementation of the resulting algorithm, and present lower bounds on whenever and .
Paper Structure (22 sections, 32 theorems, 115 equations, 6 figures, 3 algorithms)

This paper contains 22 sections, 32 theorems, 115 equations, 6 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Topological invariants from Newton pairs
  3. The polyhedral type of generic maps
  4. The polyhedral type for conical pairs
  5. Newton number of singularities
  6. Main result
  7. Affine singularities of the discriminant
  8. Boundary singularities of the discriminant
  9. Preliminaries on Newton polytopes of critical loci
  10. Preliminaries on toric varieties
  11. Orbits of a toric variety
  12. Toric morphisms
  13. Toric varieties from lattice polytopes
  14. Singularities of the discriminant at the orbits
  15. Singular points of the non-properness set
  16. ...and 7 more sections

Key Result

Theorem 1.1

The number of topological types of polynomial maps $\mathbb{C}^2\to\mathbb{C}^2\xspace$ of degree smaller or equal to $3$ and $4$ are at least $26$ and $3~217$ respectively.

Figures (6)

  • Figure 1: Some examples of convenient polytopes.
  • Figure 2: Left: The pairs $A$ (blue and orange) and $A^0$ (grey included). Right: The polytopes $\Sigma$ and $\Gamma'$ corresponding to Example \ref{['ex:main']}.
  • Figure 3: Three different edges of $A^0$ (see Example \ref{['ex:edges']}).
  • Figure 4: The halfspaces $H_{w_1,w_2}$, $H_{w_1,w_n}$ and $H_{w_{n-1},w_n}$ intersect in the red area.
  • Figure 5: Three examples of pairs $A^0$: The left-most has two dicritical long semi-origin edges that are not origin, the middle pair has one and the right-most has none.
  • ...and 1 more figures

Theorems & Definitions (85)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • ...and 75 more