# The image conjecture

- kind: conjecture
- status: refuted (evidence tier V3)
- year: 2010
- authors: W. Zhao (J. Algebra 2010)
- in one line: IC — images of commuting order-one operators are Mathieu–Zhao spaces

**Statement.** Let $A$ be a commutative algebra over a field of characteristic $0$, and let $\Phi$ be a commuting family of differential operators $\Phi_i=h_i(z)+\sum_jc_{ij}\partial_{z_j}$ on $A[z]$ with constant leading coefficients. Then $\operatorname{Im}\Phi=\sum_i\Phi_iA[z]$ is a Mathieu–Zhao subspace of $A[z]$.

A Mathieu–Zhao subspace $M$ is one with the property that if every power $f^m$ lies in $M$, then $gf^m$ lies in $M$ for every $g$ and all large $m$. Zhao proposed the image conjecture to put his vanishing conjecture and Mathieu's conjecture in one frame. He showed that the Jacobian conjecture, and hence the Dixmier conjecture, are each equivalent to special cases of it.[1] So [the image conjecture implies the Jacobian conjecture in every dimension](/wiki/Statement:jc-n.md). It also [implies the generalized vanishing conjecture, over all dimensions](/wiki/Statement:generalized-vanishing.md).[2]

It is false, in some finite dimension. No explicit instance of the image conjecture itself follows from the known counterexamples, because the implication to GVC is known only over all dimensions at once. Zhao's 2010 paper already gives a counterexample in positive characteristic (Example 2.7). Van den Essen, Wright and Zhao proved cases of it and extracted the Factorial Conjecture, whose status after 2026 the research for this page did not check.[3]

The [Gaussian moments conjecture](/wiki/Statement:gaussian-moments.md) is the image conjecture for one commuting family, the operators $\partial_i-x_i$. It fails explicitly in three variables.

## Relations

- implies [jc-n](/wiki/Statement:jc-n.md) — He showed that the Jacobian conjecture, and hence the Dixmier conjecture, are each equivalent to special cases of it. So the image conjecture implies the Jacobian conjecture in every dimension. [1]
- implies [generalized-vanishing](/wiki/Statement:generalized-vanishing.md) — It also implies the generalized vanishing conjecture, over all dimensions. [2]

## References

1. W. Zhao, [Images of commuting differential operators of order one with constant leading coefficients](https://arxiv.org/abs/0902.0210), J. Algebra 324 (2010) 231–247.
2. A. van den Essen, [An introduction to Mathieu subspaces](https://arxiv.org/abs/1907.06107), arXiv:1907.06107: IC(n) ∀n ⇒ GVC(n) ∀n ⇒ VC(n) ∀n ⟺ JC(n) ∀n.
3. A. van den Essen, D. Wright, W. Zhao, [On the Image Conjecture](https://arxiv.org/abs/1008.3962), J. Algebra 340 (2011) 211–224.

## Relations to this page

- generalizes [gaussian-moments](/wiki/Statement:gaussian-moments.md) — So GMC is the image conjecture for this one commuting family.
- related to [mathieu-conjecture](/wiki/Statement:mathieu-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.

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