The generalized vanishing conjecture

The generalized vanishing conjecture replaces the Laplacian of Zhao's vanishing conjecture 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, 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.

1. ↑ 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).
2. ↑ A. Dvorsky, An Explicit Five-Variable Counterexample to the Generalized Vanishing Conjecture, arXiv:2608.07338 (7 August 2026).