On the equivalence of the Jacobian, Dixmier and Poisson Conjectures in any characteristic

TL;DR

This paper demonstrates the equivalence of the Jacobian, Dixmier, and Poisson conjectures across all characteristics, leveraging recent advances in algebraic structures and reformulations in positive characteristic.

Contribution

It establishes the equivalence of three major conjectures in algebra in any characteristic, providing a new proof of their equivalence in characteristic zero.

Findings

Proves the equivalence of the Jacobian, Dixmier, and Poisson conjectures in any characteristic.

Provides a new proof of the equivalence of the complex versions of the Jacobian and Dixmier conjectures.

Utilizes recent results on ring homomorphisms of Azumaya algebras and reformulations in positive characteristic.

Abstract

Thanks to recent results on ring homomorphisms of Azumaya algebras and to the following ones about endomorphisms of canonical Poisson algebras and Dirac quantum algebras, and about the reformulation in positive characteristic of these conjectures in characteristic zero, we prove the equivalence of these three conjectures in any characteristic, giving also by this way a new proof of the equivalence of the complex version of the two first conjectures recently proved by Y. Tsuchimoto.