The Quartic Hessian Conjecture in Dimension Four
Zixiang Ni
math.AG
Aug 14, 2026 · v1
cs.AR
TL;DR
Lean 4 with Mathlib, alongside SymPy, was used for auxiliary checks of selected algebraic identities; these checks are not part of the logical proof.
Abstract
The Hessian conjecture asks whether a polynomial with nonzero constant Hessian determinant has a polynomial gradient inverse. It is known in dimensions at most three, false in dimensions at least five, and open in dimension four. We prove its four-variable quartic case. The top homogeneous part has zero Hessian determinant and, by the four-dimensional homogeneous Hesse theorem, is a cone. We divide its cone representative into three exhaustive types: a genuinely ternary quartic with nonzero ternary Hessian, a genuinely binary quartic, and a fourth power of a linear form. In the first type, the degree-seven determinant equation forces the cubic part to be affine-linear in the cone direction. In the binary type, the degree-six equation gives a constant null direction in a two-variable Hessian of the cubic part. In the unary type, the degree-five equation and a constant-direction lemma give the same conclusion. Every type therefore reduces to \[ f=P(x_1,x_2,x_3)+x_4Q(x_1,x_2,x_3)+a x_4^2, \qquad °Q\leq2. \] We prove, independently of the degree or top part of \(P\), that every constant-Hessian polynomial of this form has a polynomial gradient inverse. The branch \(a\ne0\) descends from the known three-dimensional Hessian conjecture after a Schur complement. When \(a=0\), an isotropic-cone rank analysis eliminates rank two, solves the rank-one exception by an explicit triangular inverse, and reduces rank zero to the two-dimensional Hessian conjecture. The coupled degree-six identity is retained throughout; no component with respect to a fixed quadratic form is separated.
Problem
The Hessian conjecture asks whether a polynomial with nonzero constant Hessian determinant has a polynomially invertible gradient. It is known for dimensions at most three, false for dimensions five and above, and open in dimension four. The quartic case in four variables was unresolved.
Approach
By the homogeneous Hesse theorem, the top quartic part is a cone. It is classified into three exhaustive types: genuinely ternary, genuinely binary, and a fourth power of a linear form. In each type, degree-seven, degree-six or degree-five determinant equations force the polynomial into the form P(x1,x2,x3)+x4·Q(x1,x2,x3)+a·x4² with deg Q ≤ 2. A uniform inversion proposition then handles this form: the a≠0 branch reduces via a Schur complement to the three-dimensional Hessian conjecture, and the a=0 branch uses an isotropic-cone rank analysis.
Results
The Hessian conjecture holds in dimension four for every potential of degree at most four. The inversion proposition also covers potentials of the reduced form with P of arbitrary degree.
