Difference between revisions
generalized-vanishing
Status and infobox
- status:
—→ refuted - tier:
—→ V3 - kind:
—→ conjecture - year:
—→ 2007 - 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 $m$. - short:
—→ GVC — Λᵐ(Pᵐ) = 0 for all m forces Λᵐ(QPᵐ) = 0 for large m
Relations
- + implies zhao-vanishing: So the JC counterexample makes GVC, which implies the vanishing conjecture in every dimension, false in some dimension.
Source +16 −0
+---
+title: The generalized vanishing conjecture
+short: GVC — Λᵐ(Pᵐ) = 0 for all m forces Λᵐ(QPᵐ) = 0 for large m
+symbol: GVC
+kind: conjecture
+status: refuted
+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
+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 [[Statement:zhao-vanishing|Zhao's vanishing conjecture]] by any constant-coefficient operator, and allows a test polynomial $Q$.<ref name="z">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).</ref> 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 [[implies::Statement:zhao-vanishing|GVC, which implies the vanishing conjecture in every dimension,]] 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).<ref>A. Dvorsky, [An Explicit Five-Variable Counterexample to the Generalized Vanishing Conjecture](https://arxiv.org/abs/2608.07338), arXiv:2608.07338 (7 August 2026).</ref> 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.
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+<references />
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Revision 255 as rendered
The generalized vanishing conjecture replaces the Laplacian of Zhao's vanishing conjecture by any constant-coefficient operator, and allows a test polynomial .[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, false in some dimension.
It fails explicitly in five variables. Dvorsky takes , and the third-order operator . Then but for every (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.
- ↑ W. Zhao, A vanishing conjecture on differential operators with constant coefficients, Acta Math. Vietnam. 32 (2007) 259–286. M. de Bondt, A few remarks on the Generalized Vanishing Conjecture, Arch. Math. (2013).
- ↑ A. Dvorsky, An Explicit Five-Variable Counterexample to the Generalized Vanishing Conjecture, arXiv:2608.07338 (7 August 2026).