Difference between revisions
mathieu-conjecture
Status and infobox
- status:
—→ refuted - tier:
—→ V3 - kind:
—→ conjecture - year:
—→ 1997 - 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 $m$. - short:
—→ On compact G: ∫fᵐ = 0 for all m forces ∫fᵐg = 0 for large m
Relations
- + implies jc-n: 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.
- + related to image-conjecture: 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.
Source +21 −0
+---
+title: The Mathieu conjecture
+short: On compact G: ∫fᵐ = 0 for all m forces ∫fᵐg = 0 for large m
+kind: conjecture
+status: refuted
+tier: V3
+year: 1997
+authors: O. Mathieu (Reims 1995; published 1997)
+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::Statement:jc-n|implies the Jacobian conjecture in every dimension]].<ref name="m">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.</ref> 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 [[related::Statement:image-conjecture|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 $N\ge3$.<ref>K. Zwart, [Mathieu's approach to the Jacobian Conjecture](https://arxiv.org/abs/2511.16561), Expo. Math. 44(3) (2026) 125766.</ref> 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).<ref>F. Santibañez-Leal, [The Collateral Impact of the Three-Dimensional Jacobian Counterexample](https://zenodo.org/records/22834179), Zenodo, 18 September 2026.</ref>
+- 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.<ref>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.</ref>
+
+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.
+
+<references />
+
Revision 270 as rendered
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 .[2] Zhang's note of 20 July draws the same conclusion for SU(3).
- A Zenodo note reads the implication as covering only maps with homogeneous. On that reading the known counterexamples give only , and 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 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.
- ↑ 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.
- ↑ K. Zwart, Mathieu's approach to the Jacobian Conjecture, Expo. Math. 44(3) (2026) 125766.
- ↑ F. Santibañez-Leal, The Collateral Impact of the Three-Dimensional Jacobian Counterexample, Zenodo, 18 September 2026.
- ↑ 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.