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PreprintCC BY 4.0arXiv

On mappings with Jacobian one

  • Zbigniew Jelonek
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Abstract

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≥ 7, then a generic element of this set is a counterexample to the Jacobian Conjecture.

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Cite

BibTeXRIS
BibTeX
@misc{jelonek2026mappings,
  title = {{On mappings with Jacobian one}},
  author = {Zbigniew Jelonek},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.20597},
  eprint = {2607.20597},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.20597},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On mappings with Jacobian one
AU  - Zbigniew Jelonek
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2607.20597
UR  - https://arxiv.org/abs/2607.20597
AB  - We show that the set $A(n, d)$ of polynomial automorphisms F : Bbb C^n → 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≥ 3, d≥ 7, then a generic element of this set is a counterexample to the Jacobian Conjecture.
ER  -