Author

Zbigniew Jelonek

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Preprint Jul 2026

On mappings with Jacobian one

We show that the set $A(n, d)$ of polynomial automorphisms $F : \Bbb C^n \to \Bbb C^n$ 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.

Zbigniew Jelonek · 0 citations

Preprint Aug 2026

On Buzzard's Theoren

Over any infinite field we prove existence of polynomial automorphism with prescribed differentials at points. More precisely, let $\K$ be an infinite field and $a_1,\ldots, a_k;$ $b_1,\ldots , b_k$ be two sequences of different points in~$\K^n$, $n\ge 2$. For any sequence $L_1,\ldots, L_k\in SL(n,\K)$ there exists a polynomial automorphism $\Phi: \K^n\to \K^n$ with jacobian one such that $\Phi(a_i)=b_i$ and $d_{a_i} \Phi= L_i$ for $i=1,\ldots, k.$

Zbigniew Jelonek, Gustavo Menani, Maria Michalska · 0 citations