Holonomic modules and 1-generation in the Jacobian Conjecture

Abstract

A polynomial endomorphism $\sigma\in {\rm End}_K(P_n)$ is called a Jacobian map if its Jacobian is a nonzero scalar (the field has zero characteristic). 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 1-generated for all Jacobian maps $\sigma$. It is shown that the $A_n$-module ${}^{\sigma} P_n$ is 1-generated 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 ${\rm deg} (\sigma )$ 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 $A_n$, the images of the Jacobian maps, endomorphisms of the Weyl algebra $A_n$ 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).