# Gradient counterexamples

- kind: counterexample
- year: 2026
- authors: F. Santibañez-Leal (48 variables); T. Prellberg (40 and 38)
- in one line: Non-injective x + ∇f, f a Hessian-nilpotent quartic, in 48, 40, 38 variables

**Statement.** There are homogeneous quartics $f$ over $\mathbb Q(i)$ with $\operatorname{Hess}f$ nilpotent whose gradient maps $X+\nabla f$ are not injective, in 48, 40 and 38 variables.

The [symmetric construction of de Bondt and van den Essen](/wiki/Statement:de-bondt-van-den-essen-symmetric.md) 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](/wiki/Statement:zhao-vanishing.md). None has been re-run by the atlas. They come from the [cubic-homogeneous counterexamples](/wiki/Statement:cubic-homogeneous-counterexamples.md) by doubling.

## Relations

- uses [de-bondt-van-den-essen-symmetric](/wiki/Statement:de-bondt-van-den-essen-symmetric.md) — 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.
- refutes [zhao-vanishing](/wiki/Statement:zhao-vanishing.md) — So these quartics are explicit counterexamples to Zhao's vanishing conjecture, where the Jacobian counterexample alone gives only existence.
- related to [cubic-homogeneous-counterexamples](/wiki/Statement:cubic-homogeneous-counterexamples.md) — They come from the cubic-homogeneous counterexamples by doubling.

## References

1. F. Santibañez-Leal, [The consequence cascade of the Jacobian counterexample](https://zenodo.org/records/21503372), Zenodo, 23 July 2026, doi:10.5281/zenodo.21503372. Unrefereed.
2. T. Prellberg, [Collision-Generated Compression for Homogeneous Keller Maps](https://arxiv.org/abs/2608.12543), arXiv:2608.12543.

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