Difference between revisions
hessian-hc4
Status and infobox
- status:
—→ open - kind:
—→ conjecture - statement:
—→ If $\varphi\in\mathbb C[x_1,x_2,x_3,x_4]$ has constant nonzero Hessian determinant, then $\nabla\varphi$ is a polynomial automorphism of $\mathbb C^4$. - short:
—→ HC₄ — the one open Hessian case; it implies JC₂
Relations
- + special case of hessian-conjecture: After July 2026, HC₄ and JC₂ are the two statements of the Jacobian and Hessian families still unsettled. HC₄ is the four-variable case of the family, so HC₄ is a special case of the Hessian conjecture in dimension n.
- + implies jc-plane: Therefore HC₄ implies the plane Jacobian conjecture JC₂.
Source +19 −0
+---
+title: HC₄: the Hessian conjecture in four variables
+short: HC₄ — the one open Hessian case; it implies JC₂
+symbol: HC₄
+kind: conjecture
+status: open
+statement: If $\varphi\in\mathbb C[x_1,x_2,x_3,x_4]$ has constant nonzero Hessian determinant, then $\nabla\varphi$ is a polynomial automorphism of $\mathbb C^4$.
+---
+After July 2026, HC₄ and JC₂ are the two statements of the Jacobian and Hessian families still unsettled.<ref name="my">G. Meng, L. Yang, [A five-variable counterexample to the Hessian conjecture, and the low-dimensional status of the Jacobian and Hessian conjectures](https://arxiv.org/abs/2607.22198), arXiv:2607.22198: “Exactly two statements of the two families remain unsettled, JC₂ and HC₄, linked by HC₄ ⇒ JC₂.”</ref> HC₄ is the four-variable case of the family, so [[special-case-of::Statement:hessian-conjecture|HC₄ is a special case of the Hessian conjecture in dimension n]].
+A plane Keller map $F$ gives $\varphi(x,y)=\langle y,F(x)\rangle$ in four variables, with Hessian determinant $(\det JF)^2$. If $\nabla\varphi$ is injective, so is $F$. Therefore [[implies::Statement:jc-plane|HC₄ implies the plane Jacobian conjecture JC₂]]. The converse is not known, so HC₄ is a stronger sufficient condition for JC₂, not a restatement of it.<ref name="my"/>
+
+Evidence so far:
+
+- Ni proves HC₄ for potentials of degree 4 (unrefereed).<ref>Z. Ni, [The Quartic Hessian Conjecture in Dimension Four](https://arxiv.org/abs/2608.14217), arXiv:2608.14217 (14 August 2026), Theorem 1.2. Unrefereed.</ref> With Wang's theorem for degree at most 3, this covers every potential of degree at most 4.
+- Hubbers (1994) proved the Jacobian conjecture for cubic-homogeneous Keller maps in dimension 4. That covers quartic potentials with no cubic part.
+- de Bondt and van den Essen proved the case $x+H$ with $JH$ symmetric and nilpotent for $n\le4$, in any degree.
+
+<references />
+
Revision 260 as rendered
After July 2026, HC₄ and JC₂ are the two statements of the Jacobian and Hessian families still unsettled.[1] HC₄ is the four-variable case of the family, so HC₄ is a special case of the Hessian conjecture in dimension n.
A plane Keller map gives in four variables, with Hessian determinant . If is injective, so is . Therefore HC₄ implies the plane Jacobian conjecture JC₂. The converse is not known, so HC₄ is a stronger sufficient condition for JC₂, not a restatement of it.[1]
Evidence so far:
- Ni proves HC₄ for potentials of degree 4 (unrefereed).[2] With Wang's theorem for degree at most 3, this covers every potential of degree at most 4.
- Hubbers (1994) proved the Jacobian conjecture for cubic-homogeneous Keller maps in dimension 4. That covers quartic potentials with no cubic part.
- de Bondt and van den Essen proved the case with symmetric and nilpotent for , in any degree.
- ↑ a b G. Meng, L. Yang, A five-variable counterexample to the Hessian conjecture, and the low-dimensional status of the Jacobian and Hessian conjectures, arXiv:2607.22198: “Exactly two statements of the two families remain unsettled, JC₂ and HC₄, linked by HC₄ ⇒ JC₂.”
- ↑ Z. Ni, The Quartic Hessian Conjecture in Dimension Four, arXiv:2608.14217 (14 August 2026), Theorem 1.2. Unrefereed.