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The Generalized Vanishing Conjecture: The Two-Variable Theorem and the First Failing Dimension

A mix of human-written and AI-generated textHuman understanding: some partsmath.AG — Algebraic Geometrymath.AC — Commutative Algebra

Contributed by Roy van Rijn

Version 2 / Oct 27, 2026 / CC BY 4.0

Abstract

We determine the exact dimensional range of the Generalized Vanishing Conjecture (GVC) for constant-coefficient differential operators over fields of characteristic zero. In two variables, we prove that Λm(Pm)=0\Lambda^m(P^m)=0 for every m≥1m\geq 1 implies Λm(QPm)=0\Lambda^m(QP^m)=0 for all sufficiently large mm and every fixed polynomial QQ. The proof combines Hall's marriage theorem, Newton polygon geometry, prime-dilation arguments, and the constant-term theorem of Duistermaat and van der Kallen. We then construct a homogeneous three-variable counterexample. Setting ρ=t2+xy\rho=t^2+xy, A=ρ+x2A=\rho+x^2, C=yρ2−2xt2ρ−x3t2C=y\rho^2-2xt^2\rho-x^3t^2, and Δ=4∂x∂y+∂t2\Delta=4\partial_x\partial_y+\partial_t^2, the pair Λ=Δ6\Lambda=\Delta^6 and P=AC2P=AC^2 satisfies Λm(Pm)=0\Lambda^m(P^m)=0 for every m≥1m\geq1, while Λm(x2Pm)≠0\Lambda^m(x^2P^m)\ne0 for every m≥1m\geq1. The construction is a homogeneous endpoint-contact lift of Long's three-dimensional Gaussian counterexample and extends to an explicit cusp-profile family with arbitrary winding, profile, and radial suspension. In particular, every Δk\Delta^k for k≥6k\geq6 admits such a failure. Consequently, unrestricted, and already homogeneous, GVC holds in dimension nn exactly when n≤2n\leq2.

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Mainly developed using OpenAI Codex, no obvious mistakes found by manual checks, but not yet checked/proof-read thoroughly.

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References

  1. de Bondt, Michiel. A Few Remarks on the Generalized Vanishing Conjecture. Archiv der Mathematik, vol. 100, no. 6, pp. 533–538. 2013DOI
  2. Duistermaat, J. J.; van der Kallen, Wilberd. Constant Terms in Powers of a Laurent Polynomial. Indagationes Mathematicae, vol. 9, no. 2, pp. 221–231. 1998DOI
  3. Feng, Zhenzhen; Sun, Xiaosong. A Case of the Generalized Vanishing Conjecture. 2018. arXiv:1801.04563arXiv
  4. Long, Christopher D. Small Counterexamples to the Gaussian Moments Conjecture. 2026. arXiv:2607.18186arXiv
  5. van den Essen, Arno; Willems, Roel; Zhao, Wenhua. Some Results on the Vanishing Conjecture of Differential Operators with Constant Coefficients. Journal of Pure and Applied Algebra, vol. 219, no. 9, pp. 3847–3861. 2015arXiv
  6. Wilson, Michael. A Face-Isolation Proof of the Two-Variable Gaussian Moments Conjecture. 2026. arXiv:2607.23887arXiv
  7. Zhao, Wenhua. A Vanishing Conjecture on Differential Operators with Constant Coefficients. Acta Mathematica Vietnamica, vol. 32, no. 3, pp. 259–286. 2007arXiv

Version history

  1. v1Submitted by Roy van RijnInitial depositSupersededOct 27, 2026
  2. v2Submitted by Roy van RijnSubmitted as a new versionCurrentOct 28, 2026