Holonomic modules and 1-generation in the Jacobian Conjecture
Let $P_n$ be a polynomial algebra in $n$ indeterminates over a field $K$ of characteristic zero. An endomorphism $\sigma \in \mathrm{End}_K(P_n)$ is called a Jacobian map if its Jacobian is a nonzero scalar. Each Jacobian map $\sigma $ is extended to an endomorphism $\sigma $ of the Weyl algebra $A_...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2024-09-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.556/ |
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| Summary: | Let $P_n$ be a polynomial algebra in $n$ indeterminates over a field $K$ of characteristic zero. An endomorphism $\sigma \in \mathrm{End}_K(P_n)$ is called a Jacobian map if its Jacobian is a nonzero scalar. Each Jacobian map $\sigma $ is extended to an endomorphism $\sigma $ of the Weyl algebra $A_n$.The Jacobian Conjecture (JC) says that every Jacobian map is an automorphism. Clearly, the Jacobian Conjecture is true iff the twisted (by $\sigma $) $P_n$-module ${}^{\sigma } P_n$ is cyclic for all Jacobian maps $\sigma $. It is shown that the $A_n$-module ${}^{\sigma } P_n$ is cyclic for all Jacobian maps $\sigma $. Furthermore, the $A_n$-module ${}^{\sigma } P_n$ is holonomic and as a result has finite length. An explicit upper bound is found for the length of the $A_n$-module ${}^{\sigma } P_n$ in terms of the degree $\deg (\sigma )$ of the Jacobian map $\sigma $. Analogous results are given for the Conjecture of Dixmier and the Poisson Conjecture. These results show that the Jacobian Conjecture, the Conjecture of Dixmier and the Poisson Conjecture are questions about holonomic modules for the Weyl algebra $A_n$ and the images of the Jacobian maps, of the endomorphisms of the Weyl algebra $A_n$ and of the Poisson endomorphisms are large in the sense that further strengthening of the results on largeness would be either to prove the conjectures or produce counter examples.A short direct algebraic (without reduction to prime characteristic) proof is given of the equivalence of the Jacobian and the Poisson Conjectures (this gives a new short proof of the equivalence of the Jacobian, Poisson and Dixmier Conjectures). |
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| ISSN: | 1778-3569 |