The symmetric reduction
JC in every dimension ⇔ JC for gradient maps x + ∇f, f a Hessian-nilpotent quartic
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 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.
- ↑ 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.
Relations
- uses The cubic reduction of Bass–Connell–Wright and Yagzhev proved · V3
The symmetric reduction turns the Jacobian conjecture into a question about gradient maps of Hessian-nilpotent quartics. The construction takes a cubic-homogeneous Keller map in n variables and produces a Hessian-nilpotent quartic in variables. [1]
- related to The Hessian conjecture in dimension n refuted · V1
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.
Relations to this page
- related to Drużkowski's reduction to cubic-linear maps proved · V3
So a counterexample can be put in cubic-linear form or in the symmetric form of de Bondt and van den Essen, but never in both at once.
- used by Gradient counterexamples
The symmetric construction of de Bondt and van den Essen turns a cubic-homogeneous Keller map in variables into a gradient map of a Hessian-nilpotent quartic in variables.
- used by Zhao's vanishing conjecture refuted · V3
By the symmetric reduction, the Jacobian conjecture reduces to gradient maps with a Hessian-nilpotent homogeneous quartic (Zhao's argument starts from the de Bondt–van den Essen symmetric form).
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