tangent-sweep

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.

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