The polyhedral type of a polynomial map on the plane
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909106134056960 |
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| author | Hilany, Boulos El Rose, Kemal |
| author_facet | Hilany, Boulos El Rose, Kemal |
| contents | Two continuous maps $f, g : \mathbb{C}^2\to\mathbb{C}^2$ are said to be topologically equivalent if there exist homeomorphisms $φ,ψ:\mathbb{C}^2\to\mathbb{C}^2$ satisfying $ψ\circ f\circφ= g$. 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 $d=4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08993 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The polyhedral type of a polynomial map on the plane Hilany, Boulos El Rose, Kemal Algebraic Geometry Combinatorics Primary 14D06, Secondary: 14T20, 58K15 Two continuous maps $f, g : \mathbb{C}^2\to\mathbb{C}^2$ are said to be topologically equivalent if there exist homeomorphisms $φ,ψ:\mathbb{C}^2\to\mathbb{C}^2$ satisfying $ψ\circ f\circφ= g$. 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 $d=4$. |
| title | The polyhedral type of a polynomial map on the plane |
| topic | Algebraic Geometry Combinatorics Primary 14D06, Secondary: 14T20, 58K15 |
| url | https://arxiv.org/abs/2402.08993 |