# The Mathieu conjecture

- kind: conjecture
- status: refuted (evidence tier V3)
- year: 1997
- authors: O. Mathieu (Reims 1995; published 1997)
- in one line: On compact G: ∫fᵐ = 0 for all m forces ∫fᵐg = 0 for large m

**Statement.** Let $G$ be a compact connected real Lie group with Haar measure, and $f,g$ complex $G$-finite functions on $G$. If $\int_Gf^m=0$ for all $m\ge1$, then $\int_Gf^mg=0$ for all large $m$.

Mathieu proved that his conjecture, for all compact connected Lie groups, [implies the Jacobian conjecture in every dimension](/wiki/Statement:jc-n.md).[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](/wiki/Statement:image-conjecture.md).

**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.

## Relations

- implies [jc-n](/wiki/Statement:jc-n.md) — Mathieu proved that his conjecture, for all compact connected Lie groups, implies the Jacobian conjecture in every dimension. The July 2026 counterexample therefore refutes it as stated. [1]
- related to [image-conjecture](/wiki/Statement:image-conjecture.md) — 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.

## References

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](https://arxiv.org/abs/2511.16561), Expo. Math. 44(3) (2026) 125766.
3. F. Santibañez-Leal, [The Collateral Impact of the Three-Dimensional Jacobian Counterexample](https://zenodo.org/records/22834179), Zenodo, 18 September 2026.
4. C. D. Long, [Counterexamples to the xz-Conjecture and the Mathieu Conjecture for SU(2)](https://arxiv.org/abs/2607.19012), arXiv:2607.19012 (21 July 2026); [The Mathieu Property for Compact Connected Lie Groups](https://arxiv.org/abs/2609.16178), arXiv:2609.16178.

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