The vanishing conjecture states that for every homogeneous polynomial 
of degree 4, 
for every 
implies 
for all sufficiently large 
, where 
is the Laplacian.
Zhao (2007) proved that the all-dimensional vanishing conjecture is equivalent to the all-dimensional Jacobian conjecture. The three-dimensional counterexample that refuted the Jacobian conjecture in 2026 therefore shows that the vanishing conjecture is false in some finite dimension, although the equivalence does not identify the smallest such dimension (Zhang 2026).
