# The generalized vanishing conjecture

- kind: conjecture
- status: refuted (evidence tier V3)
- year: 2007
- authors: W. Zhao (Acta Math. Vietnam. 2007); the form with a test polynomial Q from de Bondt and van den Essen
- in one line: GVC — Λᵐ(Pᵐ) = 0 for all m forces Λᵐ(QPᵐ) = 0 for large m

**Statement.** For a constant-coefficient differential operator $\Lambda$ on $\mathbb C[z]$ and polynomials $P,Q$: if $\Lambda^m(P^m)=0$ for all $m\ge1$, then $\Lambda^m(QP^m)=0$ for all large $m$.

The generalized vanishing conjecture replaces the Laplacian of [Zhao's vanishing conjecture](/wiki/Statement:zhao-vanishing.md) by any constant-coefficient operator, and allows a test polynomial $Q$.[1] Over all dimensions, the chain runs from the image conjecture through GVC to VC, and VC is equivalent to JC. So the JC counterexample makes [GVC, which implies the vanishing conjecture in every dimension,](/wiki/Statement:zhao-vanishing.md) false in some dimension.

It fails explicitly in five variables. Dvorsky takes $P=(t+c)(ad+bt)$, $Q=c$ and the third-order operator $\Lambda=\partial_t(\partial_a\partial_d-\partial_b\partial_c)$. Then $\Lambda^m(P^m)=0$ but $\Lambda^m(QP^m)=(-1)^m(m!)^2(m+1)!\,t\neq0$ for every $m\ge1$ (unrefereed).[2] The example does not come from a Keller map, and it is an infinite family of identities, not a finite certificate, so the atlas cannot re-check it by evaluation.

## Relations

- implies [zhao-vanishing](/wiki/Statement:zhao-vanishing.md) — So the JC counterexample makes GVC, which implies the vanishing conjecture in every dimension, false in some dimension.

## References

1. W. Zhao, [A vanishing conjecture on differential operators with constant coefficients](https://arxiv.org/abs/0704.1691), Acta Math. Vietnam. 32 (2007) 259–286. M. de Bondt, [A few remarks on the Generalized Vanishing Conjecture](https://arxiv.org/abs/1206.2836), Arch. Math. (2013).
2. A. Dvorsky, [An Explicit Five-Variable Counterexample to the Generalized Vanishing Conjecture](https://arxiv.org/abs/2608.07338), arXiv:2608.07338 (7 August 2026).

## Relations to this page

- implied by [image-conjecture](/wiki/Statement:image-conjecture.md) — It also implies the generalized vanishing conjecture, over all dimensions.
- generalizes [zhao-vanishing](/wiki/Statement:zhao-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.

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