For every polynomial f∈C[x_1,x_2,x_3,x_4] satisfying det bigl(operatornameHess(f)bigr)∈C^×, the gradient map nabla f:C^4→C^4 has a polynomial inverse; equivalently, there exists G∈C[x_1,x_2,x_3,x_4]^4 such that G circnabla f=nabla f circ G=operatornameid_C^4, where operatornameHess(f)=(fracpartial^2fpartial x_i partial x_j)_1≤ i,j≤4.

Quartic Hessian conjecture in dimension four

For every polynomial $f\in\mathbb{C}[x_1,x_2,x_3,x_4]$ satisfying $\det\bigl(\operatorname{Hess}(f)\bigr)\in\mathbb{C}^{\times}$ , the gradient map $\nabla f:\mathbb{C}^4\to\mathbb{C}^4$ has a polynomial inverse; equivalently, there exists $G\in\mathbb{C}[x_1,x_2,x_3,x_4]^4$ such that $G\circ\nabla f=\nabla f\circ G=\operatorname{id}_{\mathbb{C}^4}$ , where $\operatorname{Hess}(f)=\left(\frac{\partial^2f}{\partial x_i\partial x_j}\right)_{1\leq i,j\leq4}$ .

Progress summary

Solved

An August 2026 preprint claims to settle the four-variable problem for fourth-degree polynomials, but the broader four-variable problem remains open and the claim has not been independently verified.

The dimension-four Hessian conjecture asks whether every complex polynomial with constant nonzero Hessian determinant has an invertible polynomial gradient. Zixiang Ni’s August 2026 preprint claims this for all polynomials of degree at most $4$ , while explicitly leaving degrees at least $5$ open.

Known results

The conjecture holds in dimensions $n\leq 3$ ; the $n=2$ case is attributed to Dillen and the $n=3$ case to de Bondt.
Degree at most $2$ is immediate from the affine-gradient form.
Degree $3$ follows from Wang’s theorem on quadratic Keller maps.
It is false in dimensions $n\geq 5$ , via a five-variable degree- $14$ counterexample.

August 2026 quartic claim

Ni classifies the quartic top-degree form and reduces every case to explicit polynomial inverses or known lower-dimensional results. The preprint claims the quartic case in dimension four, but provides no independent verification or referee report.

Current status (as of August 2026): The quartic four-variable case has an unrefereed claimed proof, while the full dimension-four conjecture, including degrees at least $5$ , remains open.

Sources

Sources & referencesView supporting material

Primary source

arXiv

Additional references

The Quartic Hessian Conjecture in Dimension Four — arXiv — Zixiang Ni

algebraic-geometry polynomial-maps

3 days ago