de-bondt-van-den-essen-symmetric

The symmetric reduction turns the Jacobian conjecture into a question about gradient maps of Hessian-nilpotent quartics.^[1] The construction takes a cubic-homogeneous Keller map in n variables and produces a Hessian-nilpotent quartic in $2n$ variables. It uses non-real coefficients, and it must: over ℝ a symmetric nilpotent matrix is zero, so the class would be trivial.

It connects the conjecture to Zhao's vanishing conjecture, which is stated for Hessian-nilpotent polynomials, and to the Hessian conjecture, which is the non-homogeneous gradient analogue and is cited with it by Meng and Yang.

After July 2026 the construction produced explicit gradient counterexamples in 48, 40 and 38 variables: see gradient counterexamples. The same 2005 paper gives a six-dimensional counterexample to a dependence problem that de Bondt and van den Essen had posed in 2003.

1. ↑ M. de Bondt, A. van den Essen, A reduction of the Jacobian Conjecture to the symmetric case, Proc. Amer. Math. Soc. 133 (2005) 2201–2205, doi:10.1090/S0002-9939-05-07570-2.