Gradient counterexamples

The symmetric construction of de Bondt and van den Essen turns a cubic-homogeneous Keller map in $n$ variables into a gradient map of a Hessian-nilpotent quartic in $2n$ variables. It needs non-real coefficients.

• 48 variables. Santibañez-Leal applied the construction to Thompson's 24-variable map. The result is a quartic with 382 monomials over $\mathbb Q(i)$, whose gradient map has an exact two-point collision.^[1]
• 40 variables. The lift of MacFarlane's 20-variable restriction. Prellberg writes down the explicit quartic, with 350 monomials.
• 38 variables. The lift of the 19-variable homogenization of Van Rijn's map, with 340 monomials.^[2]

Prellberg's abstract states that Zhao's vanishing conjecture fails for the 38- and 40-variable quartics. So these quartics are explicit counterexamples to Zhao's vanishing conjecture, where the Jacobian counterexample alone gives only existence. None has been re-run by the atlas. They come from the cubic-homogeneous counterexamples by doubling.

1. ↑ F. Santibañez-Leal, The consequence cascade of the Jacobian counterexample, Zenodo, 23 July 2026, doi:10.5281/zenodo.21503372. Unrefereed.
2. ↑ T. Prellberg, Collision-Generated Compression for Homogeneous Keller Maps, arXiv:2608.12543.