On mappings with Jacobian one

Abstract

We show that the set $A(n,d)$ of polynomial automorphisms $F:{\mathbb{C}}^n\to {\mathbb{C}}^n$ of degree at most $d$ and with $Jac(F)=1$ is Zariski closed. In particular every irreducible component of the set $A(n,d)$ of polynomial mappings with Jacobian $1$ is either composed with polynomial automorphisms or (generically) with counterexamples to the Jacobian Conjecture. Moreover every such component has dimension at least $n^2-1.$ In particular if the set $X(n,d)$ is irreducible, and $n\ge 3,d\ge 6$, then a generic element of this set is a counterexample to the Jacobian Conjecture.