1. Source and the counterexample
On July 20, 2026 (UTC), Levent Alpöge posted the following polynomial map on X, crediting Akhil for posing the question and Fable for producing the example.[1] The post gives the following formula.
\[ F=(P,Q,R):\mathbb C^3\longrightarrow\mathbb C^3, \] where \[ \begin{aligned} P&=(1+xy)^3z+y^2(1+xy)(4+3xy),\\ Q&=y+3x(1+xy)^2z+3xy^2(4+3xy),\\ R&=2x-3x^2y-x^3z. \end{aligned} \]Direct calculation gives
\[ \det JF\equiv -2. \]Moreover, the three distinct points
\[ \left(0,0,-\frac14\right),\qquad \left(1,-\frac32,\frac{13}{2}\right),\qquad \left(-1,\frac32,\frac{13}{2}\right) \]are all mapped to \(\left(-\frac14,0,0\right)\). Thus \(F\) is a Keller map which is not injective. Multiplying one output coordinate by \(-1/2\) normalizes the Jacobian determinant to \(1\). Hence \(\mathrm{JC}(3)\) (the Jacobian Conjecture in dimension 3) is false, and adjoining identity coordinates gives counterexamples in every dimension \(n\ge 3\).
A short SymPy verification script can be found here verify.py.
2. The Mathieu conjecture fails for \(SU(3)\)
For a compact connected Lie group \(K\), the Mathieu conjecture considers complex finite-type functions \(f,h\) on \(K\). It asserts that
\[ \int_K f(k)^m\,dk=0\quad\text{for every }m\ge1 \]should imply
\[ \int_K f(k)^m h(k)\,dk=0\quad\text{for all sufficiently large }m. \]Mathieu's representation-theoretic argument gives the fixed-dimensional implication that the Mathieu Conjecture for \(SU(N)\) implies \(\mathrm{JC}(N)\).
A good reference is Theorem 2.2 in [2]. The original argument is due to Olivier Mathieu.[3] Since \(\mathrm{JC}(3)\) is false, we now know the Mathieu Conjecture for \(SU(3)\) is false.
Equivalently, there exist finite-type functions \(f,h\) on \(SU(3)\) for which every pure power integral vanishes, but the mixed integrals are nonzero for infinitely many powers.
3. The Gaussian Moments Conjecture fails in some dimension
Let \(X=(X_1,\dots,X_N)\) be a standard real Gaussian vector and let \(P,Q\in\mathbb C[x_1,\dots,x_N]\). The Gaussian Moments Conjecture \(\mathrm{GMC}(N)\) states that
\[ \mathbb E[P(X)^m]=0\quad\text{for every }m\ge1 \]implies
\[ \mathbb E[P(X)^mQ(X)]=0\quad\text{for all sufficiently large }m. \]Derksen, van den Essen, and Zhao proved that if \(\mathrm{GMC}(N)\) holds for every \(N\), then the Jacobian conjecture holds in every dimension.[4] Therefore, \(\mathrm{GMC}(N)\text{ is false for at least one finite }N\).
Thus, for some finite Gaussian vector, all pure polynomial moments vanish while a fixed mixed moment remains nonzero for infinitely many powers. The published theorem is global in dimension, so the present argument alone does not identify the smallest failing \(N\).
4. Zhao's Vanishing Conjecture is false
Let
\[ \Delta=\sum_{i=1}^{N}\frac{\partial^2}{\partial z_i^2} \]and let \(H(z)\) be a homogeneous polynomial of degree four. Zhao's Vanishing Conjecture states that
\[ \Delta^m(H^m)=0\quad\text{for every }m\ge1 \]should imply
\[ \Delta^m(H^{m+1})=0\quad\text{for all sufficiently large }m. \]Zhao proved that the all-dimensional Vanishing Conjecture is equivalent to the all-dimensional Jacobian conjecture; he also proved that the hypothesis above is equivalent to Hessian nilpotency of \(H\).[5] Consequently, the Vanishing Conjecture is false in some finite dimension.
In particular, there exists a quartic homogeneous Hessian-nilpotent polynomial \(H\) satisfying \(\Delta^m(H^m)=0\) for every \(m\), while \(\Delta^m(H^{m+1})\neq0\) for infinitely many \(m\). It would be very interesting to see a small explicit \(H\) from the three-dimensional Keller map.
5. The Image Conjecture is false
Let
\[ A=\mathbb C[\zeta_1,\dots,\zeta_N,z_1,\dots,z_N], \qquad D_i=\frac{\partial}{\partial z_i}-\zeta_i. \]The Image Conjecture asserts that
\[ \sum_{i=1}^{N}D_i(A) \]is a Mathieu subspace of \(A\): if every power \(a^m\) lies in this image, then \(ba^m\) should also lie in it for every fixed \(b\) and all sufficiently large \(m\).
Van den Essen, Wright, and Zhao record that the Image Conjecture implies the Vanishing Conjecture, which is equivalent to the Jacobian conjecture.[6] Hence the Image Conjecture is false in some finite dimension. Note that this immediate result is also only existential.
References
- Levent Alpöge, original X post announcing the map, July 20, 2026: x.com/__alpoge__/status/2079028340955197566.
- Kevin Zwart, “Mathieu's approach to the Jacobian Conjecture,” arXiv:2511.16561.
- Olivier Mathieu, “Some conjectures about invariant theory and their applications,” in Algèbre non commutative, groupes quantiques et invariants, Séminaires et Congrès 2, Société Mathématique de France, 1997, pp. 263–279.
- Harm Derksen, Arno van den Essen, and Wenhua Zhao, “The Gaussian Moments Conjecture and the Jacobian Conjecture,” arXiv:1506.05192; published in Israel Journal of Mathematics 219 (2017), 53–68.
- Wenhua Zhao, “Hessian Nilpotent Polynomials and the Jacobian Conjecture,” arXiv:math/0409534.
- Arno van den Essen, David Wright, and Wenhua Zhao, “On the Image Conjecture,” arXiv:1008.3962. See also Arno van den Essen, “The Amazing Image Conjecture,” arXiv:1006.5801.