Let m∈ N_+, let C^[m]= C[X_1,…,X_m], and let Delta=partial_1^2+…+partial_m^2 be the Laplace operator. Let f∈ C^[m] be homogeneous of degree 4. Vanishing Conjecture. If Delta^k(f^k)=0 for all k∈ N_+, then there exists K∈ N_+ such that Delta^k(f^k+1)=0 for all k≥ K. The paper states that Zhao proved equivalence, for all dimensions, with the Jacobian Conjecture. The individual conjecture is presented as interesting and unresolved in the supplied text.

The Vanishing Conjecture for homogeneous quartics

The conjecture asks whether the stated Laplacian vanishing condition for a homogeneous quartic forces eventual vanishing of the next power. Zhao’s work connects this question in all dimensions with the Jacobian Conjecture.

Known results

Zhao (2004) proved equivalence, in all dimensions, with the Jacobian Conjecture.
Zhao (2004) identified the hypotheses with Hessian nilpotence of the quartic.
Zhao (2007) proved the conjecture under a regular-intersection condition for the projective varieties defined by $P$ and $\sigma_2(z)=\sum_i z_i^2$ .

August 2026 counterexample claim

The preprint claims a homogeneous quartic in $40$ variables with $350$ monomials satisfying $\Delta^mP^m=0$ for every $m\ge 1$ , but with $\Delta^mP^{m+1}\ne 0$ for infinitely many $m$ . If correct, this disproves the conjecture; the supplied record contains no independent verification.

Current status (as of August 2026): The conjecture remains unverified as a theorem, with a claimed forty-variable counterexample that would disprove it if independently confirmed.

The Vanishing Conjecture for homogeneous quartics

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Progress summary

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