2026/07/22 by Zbigniew Jelonek
Mathematics · #math.AC #math.AG
We show that the set A(n, d) of polynomial automorphisms F : \Bbb Cn → \Bbb Cn 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 n2-1. In particular if the set X(n,d) is irreducible, and n≥ 3, d≥ 6, then a generic element of this set is a counterexample to the Jacobian Conjecture.