Zhao's vanishing conjecture

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). 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. The result is that the vanishing conjecture in every dimension is equivalent to the Jacobian conjecture in every dimension.^[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.^[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. The smallest failing dimension is unknown. Zhang's note of 20 July asks for a small explicit instance.^[3]

1. ↑ W. Zhao, Hessian nilpotent polynomials and the Jacobian conjecture, Trans. Amer. Math. Soc. 359 (2007) 249–274.
2. ↑ A. van den Essen, An introduction to Mathieu subspaces, arXiv:1907.06107 (2019).
3. ↑ Z. Zhang, Direct Consequences of the Three-Dimensional Counterexample to the Jacobian Conjecture, 20 July 2026.