On some questions around Berest's conjecture
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911019735973888 |
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| author | Guo, Junhu Zheglov, Alexander |
| author_facet | Guo, Junhu Zheglov, Alexander |
| contents | Let $K$ be a field of characteristic zero, let $A_1=K[x][\partial ]$ be the first Weyl algebra. In this paper we prove the following two results.
Assume there exists a non-zero polynomial $f(X,Y)\in K[X,Y]$, which has a non-trivial solution $(P,Q)\in A_{1}^{2}$ with $[P,Q]=0$, and the number of orbits under the group action of $Aut(A_1)$ on solutions of $f$ in $A_{1}^{2}$ is finite. Then the Dixmier conjecture holds, i.e $\forall φ\in End(A_{1})-\{0\}$, $φ$ is an automorphism.
Assume $φ$ is an endomorphism of monomial type (in particular, it is not an automorphism, see theorem 4.1). Then it has no non-trivial fixed point, i.e. there are no $P\in A_1$, $P\notin K$, s.t. $φ(P)=P$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_13343 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On some questions around Berest's conjecture Guo, Junhu Zheglov, Alexander Algebraic Geometry Rings and Algebras 16S32 Let $K$ be a field of characteristic zero, let $A_1=K[x][\partial ]$ be the first Weyl algebra. In this paper we prove the following two results. Assume there exists a non-zero polynomial $f(X,Y)\in K[X,Y]$, which has a non-trivial solution $(P,Q)\in A_{1}^{2}$ with $[P,Q]=0$, and the number of orbits under the group action of $Aut(A_1)$ on solutions of $f$ in $A_{1}^{2}$ is finite. Then the Dixmier conjecture holds, i.e $\forall φ\in End(A_{1})-\{0\}$, $φ$ is an automorphism. Assume $φ$ is an endomorphism of monomial type (in particular, it is not an automorphism, see theorem 4.1). Then it has no non-trivial fixed point, i.e. there are no $P\in A_1$, $P\notin K$, s.t. $φ(P)=P$. |
| title | On some questions around Berest's conjecture |
| topic | Algebraic Geometry Rings and Algebras 16S32 |
| url | https://arxiv.org/abs/2203.13343 |