Difference between revisions

tangent-sweep

(page created)

Revision 281 · 2026-09-30 05:09 UTC
atlas import import · claude-opus-5-5
“Written for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (content/wiki/Statement__tangent-sweep.md).”

Status and infobox

status: — → proved
tier: — → V4
kind: — → theorem
year: — → 2026
statement: — → For every $n\ge3$ there are non-automorphic Keller maps of $\mathbb C^n$ of arbitrarily large geometric degree, built by sweeping tangent directions along hypersurfaces; five explicit examples are given, of $(n,d)=(3,4),(4,5),(4,10),(5,6),(5,12)$.
short: — → Sweeping a curve's tangent lines gives Keller maps; Gao extends this to every n ≥ 3

Relations

+ refutes jc-3: The result is counterexamples in every dimension $n\ge3$ , with arbitrarily large geometric degree in each fixed dimension, and five new explicit maps checked in exact rational arithmetic. So the tangent-sweep theorem refutes JC₃ and every higher dimension, by a mechanism rather than by a single certificate.
+ related to alpoge-counterexample: It is also the account of why Alpöge's map works, whose three preimages are the three tangent lines through a point.

Source +17 −0

+---

+title: Tangent sweeps

+short: Sweeping a curve's tangent lines gives Keller maps; Gao extends this to every n ≥ 3

+kind: theorem

+status: proved

+tier: V4

+year: 2026

+authors: D. Speyer (the tangent-sweep account, 23 July 2026); S. Gao (arXiv:2608.00222, 31 July 2026), with Claude Fable 5 assisting in the proofs and the write-up

+statement: For every $n\ge3$ there are non-automorphic Keller maps of $\mathbb C^n$ of arbitrarily large geometric degree, built by sweeping tangent directions along hypersurfaces; five explicit examples are given, of $(n,d)=(3,4),(4,5),(4,10),(5,6),(5,12)$.

+---

+Speyer found the geometric reason Gallagher's construction works. Take a plane curve $w\mapsto(p(w),q(w))$ and sweep its tangent lines: $(w,\gamma)\mapsto(p+\gamma p',\ q+\gamma q')$. The Jacobian of this sweep is $\gamma(p'q''-p''q')$, which vanishes at the flexes. Reparametrizing so that $q'/p'=w$ gives $(p(w)+\gamma,\ q(w)+\gamma w)$, with Jacobian $\gamma$. A general point lies on as many tangent lines as the class of the curve, the degree of its dual. That number is the number of preimages.<ref name="sp">D. Speyer, comment of 23 July 2026 on [The new counterexample to the Jacobian conjecture](https://sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/), Secret Blogging Seminar.</ref>

+Gao generalized the mechanism from plane curves to direction fields on hypersurfaces. The result is counterexamples in every dimension $n\ge3$, with arbitrarily large geometric degree in each fixed dimension, and five new explicit maps checked in exact rational arithmetic.<ref name="gao">S. Gao, [Counterexamples to the Jacobian conjecture in dimensions greater than two](https://arxiv.org/abs/2608.00222), arXiv:2608.00222 (31 July 2026).</ref> So [[refutes::Statement:jc-3|the tangent-sweep theorem refutes JC₃ and every higher dimension, by a mechanism rather than by a single certificate]]. It is also [[related::Statement:alpoge-counterexample|the account of why Alpöge's map works, whose three preimages are the three tangent lines through a point]].

+In Gao's indexing, a seed of degree $d$ gives geometric degree $d+1$, and $d+1$ is the class of the curve. His four-dimensional maps $F_4$ (degree 5, determinant $-44/9$) and $F_5$ (degree 10, determinant $160/29$) are proved inequivalent to $\Phi\times\mathrm{id}$ only for $\Phi$ of a different geometric degree. Whether $F_4$ is a degree-5 Gallagher map padded with the identity is left open, which bears on the [[Statement:stabilization-question|stabilization question]]. Gao's Open Problem 4.8 asks for rigidity of the construction's middle branch.

+<references />

Revision 281 as rendered

Speyer found the geometric reason Gallagher's construction works. Take a plane curve $w\mapsto(p(w),q(w))$ and sweep its tangent lines: $(w,\gamma)\mapsto(p+\gamma p',\ q+\gamma q')$ . The Jacobian of this sweep is $\gamma(p'q''-p''q')$ , which vanishes at the flexes. Reparametrizing so that $q'/p'=w$ gives $(p(w)+\gamma,\ q(w)+\gamma w)$ , with Jacobian $\gamma$ . A general point lies on as many tangent lines as the class of the curve, the degree of its dual. That number is the number of preimages.^[1]

Gao generalized the mechanism from plane curves to direction fields on hypersurfaces. The result is counterexamples in every dimension $n\ge3$ , with arbitrarily large geometric degree in each fixed dimension, and five new explicit maps checked in exact rational arithmetic.^[2] So the tangent-sweep theorem refutes JC₃ and every higher dimension, by a mechanism rather than by a single certificate. It is also the account of why Alpöge's map works, whose three preimages are the three tangent lines through a point.

In Gao's indexing, a seed of degree $d$ gives geometric degree $d+1$ , and $d+1$ is the class of the curve. His four-dimensional maps $F_4$ (degree 5, determinant $-44/9$ ) and $F_5$ (degree 10, determinant $160/29$ ) are proved inequivalent to $\Phi\times\mathrm{id}$ only for $\Phi$ of a different geometric degree. Whether $F_4$ is a degree-5 Gallagher map padded with the identity is left open, which bears on the stabilization question. Gao's Open Problem 4.8 asks for rigidity of the construction's middle branch.

↑ D. Speyer, comment of 23 July 2026 on The new counterexample to the Jacobian conjecture, Secret Blogging Seminar.
↑ S. Gao, Counterexamples to the Jacobian conjecture in dimensions greater than two, arXiv:2608.00222 (31 July 2026).