Holonomic modules and 1-generation in the Jacobian Conjecture

Volodymyr V. Bavula

doi:10.5802/crmath.556

Research article - Algebra

Holonomic modules and 1-generation in the Jacobian Conjecture

Volodymyr V. Bavula ¹

¹ School of Mathematics and Statistics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 731-738.

Abstract
Résumé

Let $P_{n}$ be a polynomial algebra in $n$ indeterminates over a field $K$ of characteristic zero. An endomorphism $σ \in {End}_{K} (P_{n})$ is called a Jacobian map if its Jacobian is a nonzero scalar. Each Jacobian map $σ$ is extended to an endomorphism $σ$ 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 $σ$ ) $P_{n}$ -module $^{σ} P_{n}$ is cyclic for all Jacobian maps $σ$ . It is shown that the $A_{n}$ -module $^{σ} P_{n}$ is cyclic for all Jacobian maps $σ$ . Furthermore, the $A_{n}$ -module $^{σ} P_{n}$ is holonomic and as a result has finite length. An explicit upper bound is found for the length of the $A_{n}$ -module $^{σ} P_{n}$ in terms of the degree $deg (σ)$ of the Jacobian map $σ$ . 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).

Soit $P_{n}$ un polynôme à $n$ indéterminées sur un corps $K$ de caractéristique zéro. Un endomorphisme $σ \in {End}_{K} (P_{n})$ est appelé une application jacobienne si son jacobien est un scalaire non nul. Chaque application jacobienne $σ$ est étendue en un endomorphisme $σ$ de l’algèbre de Weyl $A_{n}$ .

La Conjecture Jacobienne (CJ) affirme que chaque application jacobienne est un automorphisme. Clairement, la Conjecture Jacobienne est vraie si le module tordu (par $σ$ ) $^{σ} P_{n}$ est cyclique pour toutes les applications jacobienne $σ$ . Il est démontré que le module $A_{n}$ $^{σ} P_{n}$ est cyclique pour toutes les applications jacobienne $σ$ . De plus, le module $A_{n}$ - $^{σ} P_{n}$ est holonomique et, par conséquent, de longueur finie. Une borne supérieure explicite est trouvée pour la longueur du module $A_{n}$ - $^{σ} P_{n}$ en fonction du degré $deg (σ)$ de l’application jacobienne $σ$ . Des résultats analogues sont donnés pour la Conjecture de Dixmier et la Conjecture Poisson. Ces résultats montrent que la Conjecture Jacobienne, la Conjecture de Dixmier et la Conjecture Poisson sont des questions sur les modules holonomiques pour l’algèbre de Weyl $A_{n}$ et que les images des applications jacobienne, des endomorphismes de l’algèbre de Weyl $A_{n}$ et des endomorphismes de Poisson sont importantes au sens où des renforcements supplémentaires des résultats sur l’importance consisteraient soit à prouver les conjectures, soit à produire des contre-exemples.

Une démonstration directe et brève (sans réduction à la caractéristique première) est donnée de l’équivalence des Conjectures Jacobienne et Poisson (ceci donne une nouvelle démonstration brève de l’équivalence des Conjectures Jacobienne, Poisson et Dixmier).

Received: 2022-10-08
Revised: 2023-08-23
Accepted: 2023-08-23
Published online: 2024-09-17

DOI: 10.5802/crmath.556

Classification: 14R15, 14R10, 13F20, 16S32, 14F10, 16D30, 16D60, 16P90
Keywords: The Jacobian Conjecture, the Conjecture of Dixmier, the Weyl algebra, the holonomic module, the endomorphism algebra, the length, the multiplicity
Mots-clés : La conjecture jacobienne, la conjecture de Dixmier, l’algèbre de Weyl, le module holonomique, l’algèbre des endomorphismes, la longueur, la multiplicité

Author's affiliations:

Volodymyr V. Bavula ¹

¹ School of Mathematics and Statistics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Volodymyr V. Bavula. Holonomic modules and 1-generation in the Jacobian Conjecture. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 731-738. doi : 10.5802/crmath.556. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.556/

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