Counterexamples to the Jacobian conjecture in dimensions greater than two
Abstract
The Jacobian conjecture, open since 1939, asks whether every polynomial map of Cn whose Jacobian determinant is a nonzero constant must have a polynomial inverse. It was refuted in dimension three by Alpöge on July 19, 2026, with an infinite family by Gallagher (July 20) and a geometric explanation by Speyer (July 23): the counterexample sweeps the tangent lines of a plane curve. We give a self-contained account of this tangent-sweep mechanism and generalize it from plane curves to direction fields on hypersurfaces. The resulting construction, including stabilization of the three-dimensional family, produces counterexamples in every dimension greater than two and, in each dimension, of arbitrarily large geometric degree (the number of preimages of a typical point). We work it out in five explicit maps: one three-dimensional of degree four, two four-dimensional of degrees five and ten, and two five-dimensional of degrees six and twelve. We show that our explicit examples in dimensions n≥4 are not polynomially left-right equivalent to the counterexamples of Alpöge and Gallagher when these are embedded in Cn by adjoining identity coordinates. The counterexamples provide explicit examples of étale morphisms Cn→Cn that are not proper: they are everywhere unramified, and fail to be injective only through points escaping to infinity. All identities were verified in exact rational arithmetic. Part~\ref{part:equivalence} proves the higher-dimensional inequivalence results, and Appendix A determines exact fiber structures for the three-dimensional examples. A complete classification of the fibers of F4,F5,F6,F7 remains open.
Related
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- Poincaré compactifications of pullbacks of parallel vector fields under counterexamples to the real Jacobian conjecture
- Semifields in prime dimensions and counterexamples to Kaplansky's conjecture
- An elementary way to find a counterexample to the Jacobian Conjecture
- Some arithmetical aspects of the two dimensional Jacobian Conjecture
BibTeX
@misc{gao2026counterexamples,
title = {{Counterexamples to the Jacobian conjecture in dimensions greater than two}},
author = {Shuhong Gao},
howpublished = {arXiv},
year = {2026},
month = jul,
doi = {10.48550/arxiv.2608.00222},
eprint = {2608.00222},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.00222},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Counterexamples to the Jacobian conjecture in dimensions greater than two
AU - Shuhong Gao
PY - 2026
DA - 2026/07/31
PB - arXiv
DO - 10.48550/arxiv.2608.00222
UR - https://arxiv.org/abs/2608.00222
AB - The Jacobian conjecture, open since 1939, asks whether every polynomial map of C^n whose Jacobian determinant is a nonzero constant must have a polynomial inverse. It was refuted in dimension three by Alpöge on July 19, 2026, with an infinite family by Gallagher (July 20) and a geometric explanation by Speyer (July 23): the counterexample sweeps the tangent lines of a plane curve. We give a self-contained account of this tangent-sweep mechanism and generalize it from plane curves to direction fields on hypersurfaces. The resulting construction, including stabilization of the three-dimensional family, produces counterexamples in every dimension greater than two and, in each dimension, of arbitrarily large geometric degree (the number of preimages of a typical point). We work it out in five explicit maps: one three-dimensional of degree four, two four-dimensional of degrees five and ten, and two five-dimensional of degrees six and twelve. We show that our explicit examples in dimensions n≥4 are not polynomially left-right equivalent to the counterexamples of Alpöge and Gallagher when these are embedded in C^n by adjoining identity coordinates. The counterexamples provide explicit examples of étale morphisms C^n→C^n that are not proper: they are everywhere unramified, and fail to be injective only through points escaping to infinity. All identities were verified in exact rational arithmetic. Part~\ref{part:equivalence} proves the higher-dimensional inequivalence results, and Appendix A determines exact fiber structures for the three-dimensional examples. A complete classification of the fibers of F_4,F_5,F_6,F_7 remains open.
SN - 2331-8422
ER -