Automorphism Groups of Commuting Polynomial Maps of the Affine Plane
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916445444636672 |
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| author | Silverman, Jospeh H. |
| author_facet | Silverman, Jospeh H. |
| contents | Let $\mathcal{L}$ be a finite-dimensional semisimple Lie algebra of rank $N$ over an algebraically closed field of characteristic $0$. Associated to $\mathcal{L}$ is a family of polynomial folding maps $$\textsf{F}_{n}:\mathbb{A}^N\to\mathbb{A}^N\quad\text{for}\quad n\ge1$$ having the property that $\textsf{F}_{n}$ has topological degree $n^N$ and $$\textsf{F}_{m}\circ\textsf{F}_{n}=\textsf{F}_{n}\circ\textsf{F}_{m}\quad\text{for all}\quad m,n\ge1.$$ We derive formulas for the leading terms of the folding maps on $\mathbb{A}^2$ associated to the Lie algebras $\mathcal{A}_2$, $\mathcal{B}_2$, and $\mathcal{G}_2$, and we use these formulas to compute the affine automorphism group of each folding map. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10598 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Automorphism Groups of Commuting Polynomial Maps of the Affine Plane Silverman, Jospeh H. Dynamical Systems Algebraic Geometry Number Theory Primary: 37P05, Secondary: 32H50, 37F10, 37J37, 37J70 Let $\mathcal{L}$ be a finite-dimensional semisimple Lie algebra of rank $N$ over an algebraically closed field of characteristic $0$. Associated to $\mathcal{L}$ is a family of polynomial folding maps $$\textsf{F}_{n}:\mathbb{A}^N\to\mathbb{A}^N\quad\text{for}\quad n\ge1$$ having the property that $\textsf{F}_{n}$ has topological degree $n^N$ and $$\textsf{F}_{m}\circ\textsf{F}_{n}=\textsf{F}_{n}\circ\textsf{F}_{m}\quad\text{for all}\quad m,n\ge1.$$ We derive formulas for the leading terms of the folding maps on $\mathbb{A}^2$ associated to the Lie algebras $\mathcal{A}_2$, $\mathcal{B}_2$, and $\mathcal{G}_2$, and we use these formulas to compute the affine automorphism group of each folding map. |
| title | Automorphism Groups of Commuting Polynomial Maps of the Affine Plane |
| topic | Dynamical Systems Algebraic Geometry Number Theory Primary: 37P05, Secondary: 32H50, 37F10, 37J37, 37J70 |
| url | https://arxiv.org/abs/2410.10598 |