Holonomic modules and 1-generation in the Jacobian Conjecture

Let Pn be a polynomial algebra in n indeterminates over a field K of characteristic zero. An endomorphism σ ∈ EndK (Pn) 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 An. The Jacobian Conjecture (JC) says that every Jacobian map is an automorphism. Clearly, the Jacobian Conjecture is true if the twisted (by σ) Pn-module σPn is cyclic for all Jacobian maps σ. It is shown that the An-module σPn is cyclic for all Jacobian maps σ. Furthermore, the An-module σPn is holonomic and as a result has finite length. An explicit upper bound is found for the length of the An-module σPn 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 An and the images of the Jacobian maps, of the endomorphisms of the Weyl algebra An 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).