gallagher-families

The day after Alpöge's announcement, Gallagher showed that the map is the first member of a one-parameter family, not an isolated accident.^[1] Each member is a Keller map of ℂ³ that is not injective, so each member of the family refutes JC₃ independently of Alpöge's map.

The construction

Fix $c\ne0$ and a one-variable seed polynomial $p$ with $p(0)=0$, $p(1)=-c$ and $\int_0^1p=0$. Put $v=xy$, $t=x^2z$, $u=1+v$, $\gamma=1+av+bt$ and $w=u\gamma$. Then set $\beta=c+p(w)/\gamma$ and $\alpha=u+q(w)/\gamma^2$, with $a$ and $q$ determined by $p$. The map $F_p=(\alpha/x^2,\ \beta/x,\ x\gamma)$ is a polynomial map with $\det JF_p=bc$, and its generic fiber has $\deg p+1$ points.^[2] The seed $p(w)=2w-3w^2$ gives Alpöge's map. For every $n\ge3$ take $b=c=1$, $d=n-1$ and $$p_d(w)=2w-3w^2+w(1-w)\Big(w^{d-2}-\tfrac{6}{d(d+1)}\Big).$$ The three seed conditions hold because $\int_0^1w(1-w)w^{d-2}\,dw=\tfrac1{d(d+1)}$.

The mechanism is a one-variable equation of degree $\deg p+1$. Away from its discriminant it has that many distinct roots, and each root gives back a distinct input.

Evidence

• The uniform proof, for every $n$, is written but unrefereed: V4.
• The members $n=3,\dots,100$ are published with exact-arithmetic verifiers on jacobianfun.org: 98 maps with 5,047 rational preimages.^[3] The atlas has not re-run them.
• The atlas does check the determinant of the fiber-degree-4 member, as transcribed in the Fong–Fable ledger:

The map $G$ highlighted on jacobianfun.org is a different member. It has fiber degree 4, component degrees (12, 11, 4) and determinant $-6$, and it is the second point plotted on the JC₃ chart.

What it settled

The family has every prime $p\ge3$ as a fiber degree in dimension three, so the prime-degree obstruction claimed for the plane does not extend to ℂ³. Together with the exclusion of degree 2, it gives the degree spectrum of ℂ³.

It also refuted, on the day, a guess recorded in the Fong–Fable ledger: that étale degrees in ℂ³ are powers of 3. The guess was proposed by the model in that session. It was briefly given an eponym, which was withdrawn at Fong's request, because she never stated it.^[4]

1. ↑ A. Gallagher, An infinite family of counterexamples to the Jacobian Conjecture in dimension three, Zenodo preprint, 20 July 2026.
2. ↑ A. Gallagher, jacobianfun RESEARCH.md, §3 and §5, 20 July 2026.
3. ↑ jacobianfun.org counterexample atlas.
4. ↑ D. Fong and Claude Fable 5, claims ledger, corrections and naming correction of 20 July 2026.