Simple derivations and the isotropy groups

Abstract

In the paper, we study the isotropy groups of $K[x]$ if D is a simple derivation. We first study the derivation of the form: $D={\partial }_{x_1}+{\sum }_{i=2}^n(a_i{x}_i+b_i){\partial }_{x_i}$ with $a_i,b_i\in K[x_1,\dots ,x_{i-1}]$ for all $2\le i\le n$. We prove that $\text{Aut}{(K[x])}_D=\left\{id\right\}$ if D is simple with ${\mathrm{deg}}_{x_{i-1}}{a}_i\ge 1$ for all $2\le i\le n$ or $a_i\in K^{*}$ for all $3\le i\le n$ or $0\le {\mathrm{deg}}_{x_j}{b}_i\le {\mathrm{deg}}_{x_j}{a}_i$ for all $2\le j\le i-1,3\le i\le n$. Thus, we conjecture that $\text{Aut}{(K[x])}_D=\left\{id\right\}$ if D is simple with ${\prod }_{i=2}^n{a}_i\ne 0,a_i,b_i\in K[x_1,\dots ,x_{i-1}]$ for all $2\le i\le n$. Then we prove that $\text{Aut}{(K[x])}_D=\{(x_1,\dots ,x_{n-1},x_n+c)|c\in K\}$ if $D=(x_1^s{x}_2+g){\partial }_{x_1}+x_1^{s-1}{\partial }_{x_2}+g_3{\partial }_{x_3}+\cdots +g_n{\partial }_{x_n}$ or $D=(x_1^s{x}_2^t+c){\partial }_{x_1}+x_1^r{\partial }_{x_2}+g_3{\partial }_{x_3}+\cdots +g_n{\partial }_{x_n}$ with $g\in K[x_1],\mathrm{deg}(g)\le s,g(0)\ne 0$ and $g_i\in K[x_{i-1}]\backslash K$ for all $3\le i\le n$.

KEYWORDS:

• Isotropy groups
• polynomial automorphisms
• simple derivations

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13N15
• 14R10
• 13P05

Acknowledgments

The authors are very grateful to the referee for some useful comments and suggestions.

Disclosure statement

We have no relevant financial or non-financial competing interests.

Additional information

Funding

The author is supported by the NSF of Hunan Province (Grant No. 2023JJ30386), the NSF of China (Grant No. 12371020), the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 21A0056) and the Construct Program of the Key Discipline in Hunan Province.