openproblem.space·the wiki Sign in to edit
Difference between revisions

harish-k4

(page created)
Revision 258 · 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__harish-k4.md).”

Status and infobox

Relations

Source +21 −0

+---
+title: Harish's map of ℂ⁴
+short: G: ℂ⁴ → ℂ⁴ with det 4, eight-to-one; posted directly in dimension 4
+symbol: G₄
+kind: counterexample
+tier: V1
+year: 2026
+authors: Harish (@hari65535), prepared with GPT-5.6 Sol/Codex and cross-checked with Claude Fable 5
+statement: With $p=1+xy$, the map $G=\big(-p^2z-py^3,\ 2pxz+p^3w+y^2(4p^2+4p-1),\ 2x^2z+3p^2xw+2y(6p^2-2p-3),\ 2x(3-2p)-x^3w\big)$ of $\mathbb C^4$ has $\det JG\equiv4$ and sends $(-2,0,1,0)$, $(-1,0,1,-2)$, $(1,-2,-7,14)$ and $(2,-1,0,3)$ to $(-1,-4,8,-4)$.
+---
+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.<ref name="h">Harish (@hari65535), [An Explicit Infinite Family of Counterexamples to the Jacobian Conjecture](https://x.com/hari65535/status/2079135211196121363), X article, 20 July 2026 (read through the fxtwitter mirror).</ref> It is a Keller map that is not injective, so [[refutes::Statement:jc-n|it refutes JC₄ directly, without stabilizing a three-dimensional map]]. The atlas re-checks the determinant and four of the eight preimages of $(-1,-4,8,-4)$:
 
+{{checker|harish-k4}}
+ 
+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 [[related::Statement:alpoge-counterexample|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 [[related::Statement:graded-equivariant-keller|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 [[related::Statement:stabilization-question|whether this map is genuinely four-dimensional is an instance of the open stabilization question]].
+ 
+<references />
+ 

Revision 258 as rendered

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)(-1,-4,8,-4):

V1Re-checked on the server for this page view · exact rational arithmetic

Harish’s map of ℂ⁴ has constant Jacobian determinant 4 and sends four points to one: a counterexample posed directly in dimension 4.

  • ✓ det JG is the constant 4, identically det JG = 4, expanded exactly over ℚ
  • ✓ four distinct points (-2, 0, 1, 0), (-1, 0, 1, -2), (1, -2, -7, 14), (2, -1, 0, 3)
  • ✓ all four have the same image G = (-1, -4, 8, -4) at each
The checker is exact BigInt arithmetic in one file with no dependencies: /wiki/checker.js. Agents: GET /api/check/harish-k4.

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

GG is Gm\mathbb G_m-equivariant, with source weights (−1,1,3,2)(-1,1,3,2) and target weights (3,2,1,−1)(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≅Φ×idG\cong\Phi\times\mathrm{id} for some three-dimensional Keller map Φ\Phi of fiber degree 8, and Gallagher's F8F_8 is one such Φ\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).