Difference between revisions
gradient-keller-counterexamples
Status and infobox
- tier:
—→ V4 - kind:
—→ counterexample - year:
—→ 2026 - 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. - short:
—→ Non-injective x + ∇f, f a Hessian-nilpotent quartic, in 48, 40, 38 variables
Relations
- + uses de-bondt-van-den-essen-symmetric: 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.
- + refutes zhao-vanishing: So these quartics are explicit counterexamples to Zhao's vanishing conjecture, where the Jacobian counterexample alone gives only existence.
- + related to cubic-homogeneous-counterexamples: They come from the cubic-homogeneous counterexamples by doubling.
Source +18 −0
+---
+title: Gradient counterexamples
+short: Non-injective x + ∇f, f a Hessian-nilpotent quartic, in 48, 40, 38 variables
+kind: counterexample
+tier: V4
+year: 2026
+authors: F. Santibañez-Leal (48 variables); T. Prellberg (40 and 38)
+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 [[uses::Statement:de-bondt-van-den-essen-symmetric|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.<ref>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.</ref>
+- **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.<ref name="pr">T. Prellberg, [Collision-Generated Compression for Homogeneous Keller Maps](https://arxiv.org/abs/2608.12543), arXiv:2608.12543.</ref>
+
+Prellberg's abstract states that Zhao's vanishing conjecture fails for the 38- and 40-variable quartics. So [[refutes::Statement:zhao-vanishing|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 [[related::Statement:cubic-homogeneous-counterexamples|cubic-homogeneous counterexamples]] by doubling.
+
+<references />
+
Revision 257 as rendered
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. 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 , 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.
- ↑ F. Santibañez-Leal, The consequence cascade of the Jacobian counterexample, Zenodo, 23 July 2026, doi:10.5281/zenodo.21503372. Unrefereed.
- ↑ T. Prellberg, Collision-Generated Compression for Homogeneous Keller Maps, arXiv:2608.12543.