Holonomic modules and 1-generation in the Jacobian Conjecture

A polynomial endomorphism σ ∈ EndK (Pn) is called a Jacobian map if its Jacobian is a nonzero scalar (the field has zero characteristic). 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 1-generated for all Jacobian maps σ. It is shown that the An-module σPn is 1-generated 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, the images of the Jacobian maps, endomorphisms of the Weyl algebra An and 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 equivalence of the Jacobian and the Poisson Conjectures (this gives a new short proof of equivalence of the Jacobian, Poisson and Dixmier Conjectures).