On polynomial automorphisms commuting with a simple derivation
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arXiv
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| Format: | Preprint |
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2024
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| author | Montagard, Pierre-Louis Pan, Iván Rittatore, Alvaro |
| author_facet | Montagard, Pierre-Louis Pan, Iván Rittatore, Alvaro |
| contents | Let $D$ be a simple derivation of the polynomial ring $\mathbb{k}[x_1,\dots,x_n]$, where $\mathbb{k}$ is an algebraically closed field of characteristic zero, and denote by $\operatorname{Aut}(D)\subset\operatorname{Aut}(\mathbb{k}[x_1,\dots,x_n])$ the subgroup of $\mathbb{k}$-automorphisms commuting with $D$. We show that the connected component of $\operatorname{Aut}(D)$ passing through the identity is a unipotent algebraic group of dimension at most $n-2$, this bound being sharp. Moreover, $\operatorname{Aut}(D)$ is an algebraic group if and only if it is a connected ind-group. Given a simple derivation $D$, we characterize when $\operatorname{Aut}(D)$ contains a normal subgroup of translations. As an application of our techniques we show that if $n=3$, then either $\operatorname{Aut}(D)$ is a discrete group or it is isomorphic to the additive group acting by translations, and give some insight on the case $n=4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_09519 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On polynomial automorphisms commuting with a simple derivation Montagard, Pierre-Louis Pan, Iván Rittatore, Alvaro Algebraic Geometry Commutative Algebra 13N15, 14R10 Let $D$ be a simple derivation of the polynomial ring $\mathbb{k}[x_1,\dots,x_n]$, where $\mathbb{k}$ is an algebraically closed field of characteristic zero, and denote by $\operatorname{Aut}(D)\subset\operatorname{Aut}(\mathbb{k}[x_1,\dots,x_n])$ the subgroup of $\mathbb{k}$-automorphisms commuting with $D$. We show that the connected component of $\operatorname{Aut}(D)$ passing through the identity is a unipotent algebraic group of dimension at most $n-2$, this bound being sharp. Moreover, $\operatorname{Aut}(D)$ is an algebraic group if and only if it is a connected ind-group. Given a simple derivation $D$, we characterize when $\operatorname{Aut}(D)$ contains a normal subgroup of translations. As an application of our techniques we show that if $n=3$, then either $\operatorname{Aut}(D)$ is a discrete group or it is isomorphic to the additive group acting by translations, and give some insight on the case $n=4$. |
| title | On polynomial automorphisms commuting with a simple derivation |
| topic | Algebraic Geometry Commutative Algebra 13N15, 14R10 |
| url | https://arxiv.org/abs/2412.09519 |