The Mathieu conjecture

Mathieu proved that his conjecture, for all compact connected Lie groups, implies the Jacobian conjecture in every dimension.^[1] The July 2026 counterexample therefore refutes it as stated. In Zhao's language it says that the kernel of Haar integration is a Mathieu–Zhao subspace, which puts it in the same frame as the image conjecture.

Group by group, the record is less settled, and the atlas records the disagreement.

• Zwart's exposition states the fixed-dimension form “Mathieu for SU(N) implies JC on ℂᴺ” (Theorem 2.2). That kills SU(N) for every $N\ge3$.^[2] Zhang's note of 20 July draws the same conclusion for SU(3).
• A Zenodo note reads the implication as covering only maps $x-h$ with $h$ homogeneous. On that reading the known counterexamples give only $N\ge24$, and $3\le N\le23$ is open (unrefereed).^[3]
• Independently of the Keller map, Long gives an explicit counterexample on SU(2). He then claims that the property holds exactly when $G$ is a torus, so it fails for every nonabelian group, which would make the dimension question moot. Both preprints are unrefereed.^[4]

The research behind this page could not read the proof section of Zwart's exposition, so it cannot say which reading is right. The torus case is a theorem: constant terms of powers of Laurent polynomials (Duistermaat and van der Kallen, 1998). It did not fall.

1. ↑ O. Mathieu, Some conjectures about invariant theory and their applications, in Algèbre non commutative, groupes quantiques et invariants (Reims, 1995), Sémin. Congr. 2, SMF, 1997, 263–279.
2. ↑ K. Zwart, Mathieu's approach to the Jacobian Conjecture, Expo. Math. 44(3) (2026) 125766.
3. ↑ F. Santibañez-Leal, The Collateral Impact of the Three-Dimensional Jacobian Counterexample, Zenodo, 18 September 2026.
4. ↑ C. D. Long, Counterexamples to the xz-Conjecture and the Mathieu Conjecture for SU(2), arXiv:2607.19012 (21 July 2026); The Mathieu Property for Compact Connected Lie Groups, arXiv:2609.16178.