# Zhao's vanishing conjecture

- kind: conjecture
- status: refuted (evidence tier V3)
- year: 2007
- authors: W. Zhao (arXiv 2004; Trans. AMS 2007)
- in one line: VC — for Hessian-nilpotent quartic P, Δᵐ Pᵐ⁺¹ = 0 for large m

**Statement.** Let $\Delta$ be the Laplacian on $\mathbb C[z_1,\dots,z_n]$. If $P$ is homogeneous of degree 4 and $\Delta^mP^m=0$ for all $m\ge1$ (equivalently, $P$ is Hessian nilpotent), then $\Delta^mP^{m+1}=0$ for all large $m$.

By the symmetric reduction, the Jacobian conjecture reduces to gradient maps $z-\nabla P$ with $P$ a Hessian-nilpotent homogeneous quartic ([Zhao's argument starts from the de Bondt–van den Essen symmetric form](/wiki/Statement:de-bondt-van-den-essen-symmetric.md)). Zhao wrote the formal inverse of such a map as $z+\nabla Q$, where $Q$ is a series in the terms $\Delta^mP^{m+1}$. So [the map has a polynomial inverse exactly when those terms vanish for large m, which is a statement about the formal inverse](/wiki/Statement:formal-inverse.md). The result is that [the vanishing conjecture in every dimension is equivalent to the Jacobian conjecture in every dimension](/wiki/Statement:jc-n.md).[1] The equivalence does not preserve the dimension, because it passes through the cubic-homogeneous and symmetric reductions.

It is VC for the Laplacian and $Q=P$; the general form allows other constant-coefficient operators and test polynomials, so [VC is a special case of the generalized vanishing conjecture](/wiki/Statement:generalized-vanishing.md).[2]

**It is false.** Through the equivalence, the July 2026 counterexample makes VC fail in some finite dimension. Explicit failures came in August: Prellberg states that it fails for the 40- and 38-variable [gradient counterexamples](/wiki/Statement:gradient-keller-counterexamples.md). The smallest failing dimension is unknown. Zhang's note of 20 July asks for a small explicit instance.[3]

## Relations

- uses [de-bondt-van-den-essen-symmetric](/wiki/Statement:de-bondt-van-den-essen-symmetric.md) — By the symmetric reduction, the Jacobian conjecture reduces to gradient maps $z-\nabla P$ with $P$ a Hessian-nilpotent homogeneous quartic (Zhao's argument starts from the de Bondt–van den Essen symmetric form).
- uses [formal-inverse](/wiki/Statement:formal-inverse.md) — So the map has a polynomial inverse exactly when those terms vanish for large m, which is a statement about the formal inverse.
- equivalent to [jc-n](/wiki/Statement:jc-n.md) — The result is that the vanishing conjecture in every dimension is equivalent to the Jacobian conjecture in every dimension. The equivalence does not preserve the dimension, because it passes through the cubic-homogeneous and symmetric reductions. [1]
- special case of [generalized-vanishing](/wiki/Statement:generalized-vanishing.md) — It is VC for the Laplacian and $Q=P$; the general form allows other constant-coefficient operators and test polynomials, so VC is a special case of the generalized vanishing conjecture. [2]

## References

1. W. Zhao, [Hessian nilpotent polynomials and the Jacobian conjecture](https://arxiv.org/abs/math/0409534), Trans. Amer. Math. Soc. 359 (2007) 249–274.
2. A. van den Essen, [An introduction to Mathieu subspaces](https://arxiv.org/abs/1907.06107), arXiv:1907.06107 (2019).
3. Z. Zhang, [Direct Consequences of the Three-Dimensional Counterexample to the Jacobian Conjecture](https://zzhang-iu.github.io/papers/direct-consequences-jacobian/), 20 July 2026.

## Relations to this page

- implied by [generalized-vanishing](/wiki/Statement:generalized-vanishing.md) — So the JC counterexample makes GVC, which implies the vanishing conjecture in every dimension, false in some dimension.
- refuted by [gradient-keller-counterexamples](/wiki/Statement:gradient-keller-counterexamples.md) — So these quartics are explicit counterexamples to Zhao's vanishing conjecture, where the Jacobian counterexample alone gives only existence.

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