Harish's map of ℂ⁴

Harish's map was posted on 20 July 2026 at 09:23 UTC. It is the first counterexample written directly in dimension four, rather than obtained by padding a map of ℂ³.^[1] It is a Keller map that is not injective, so it refutes JC₄ directly, without stabilizing a three-dimensional map. The atlas re-checks the determinant and four of the eight preimages of $(-1,-4,8,-4)$:

The map is the integer-scaled $n=4$ member of a family $K_n:\mathbb C^n\to\mathbb C^n$, $n\ge3$, with determinant $(n-2)/2$ and generic fiber degree $n(n-2)$. The count is $n$ roots of a degree-$n$ polynomial, times $n-2$ solutions of $a^{n-2}=P'(t)$. The $n=3$ member has exactly Alpöge's components, so this family extends Alpöge's map in the dimension rather than in the degree.

$G$ is $\mathbb G_m$-equivariant, with source weights $(-1,1,3,2)$ and target weights $(3,2,1,-1)$. The weights are not in the source; the research for this page derived them from the monomials. The target weights are a permutation of the source weights, as Shaska's theorem requires of every graded Keller map, and they have mixed signs, as it requires of every graded counterexample.

Is it native? Fiber degree 8 rules out Alpöge's map padded with an identity coordinate, which has degree 3. It does not rule out $G\cong\Phi\times\mathrm{id}$ for some three-dimensional Keller map $\Phi$ of fiber degree 8, and Gallagher's $F_8$ is one such $\Phi$. So whether this map is genuinely four-dimensional is an instance of the open stabilization question.

1. ↑ Harish (@hari65535), An Explicit Infinite Family of Counterexamples to the Jacobian Conjecture, X article, 20 July 2026 (read through the fxtwitter mirror).