math.AG — Algebraic Geometry

The four-dimensional Hessian conjecture for potentials of degree at most five

Contributed by David Niedbala Giraudin

The Hessian conjecture HC_n asserts that if f in C[x_1,...,x_n] has det Hess(f) in C^×, then ∇f is a polynomial automorphism. It is known for n ≤ 3, false for n ≥ 5, and open for n = 4, where it implies the plane Jacobian conjecture. Ni proved HC_4 for potentials of degree at most four, and the author treated degree five when the top form depends on exactly three linear forms (hexagon:2611.00134). We prove HC_4 for every potential of degree at most five. The remaining top forms are a binary form or a fifth power of a linear form. Both cases are reduced, by weighted layers of the Hessian determinant and exact computer algebra over Q, to four elementary mechanisms: a Schur complement in one variable (which reduces to HC_3), a partial Legendre transform in two variables (which reduces to HC_2), a "dead direction" lemma (which reduces to HC_2 and the plane Jacobian conjecture in degrees at most (3,4), known by Moh), and explicit triangular inverses. Five self-contained verifiers are supplied as ancillary files. The paper has not been refereed.

math.AG — Algebraic Geometry

Equivalence of the Four-Variable Hessian and Planar Jacobian Conjectures

Contributed by Qiyue Tang

We prove that the four-variable Hessian conjecture and the planar Jacobian conjecture are equivalent over the complex numbers.

math.AG — Algebraic Geometry

The four-dimensional Hessian conjecture for quintic potentials with a ternary cone top

Contributed by David Niedbala Giraudin

The Hessian conjecture $\mathrm{HC}_n$ asserts that if $f\in\mathbb{C}[x_1,\dots,x_n]$ has $\det\operatorname{Hess}(f)\in\mathbb{C}^\times$, then $\nabla f$ is a polynomial automorphism. It is known for $n\le 3$, false for $n\ge 5$, and open for $n=4$, where it implies the plane Jacobian conjecture. Z. Ni proved $\mathrm{HC}_4$ for potentials of degree at most four. We prove $\mathrm{HC}_4$ for every potential of degree five whose top homogeneous form depends on exactly three linear forms. The proof combines a weighted analysis of the Hessian determinant with exact computer-algebra certificates (Groebner bases and Rabinowitsch tests over $\mathbb{Q}$); one branch reduces to plane Jacobian pairs of degrees $(4,3)$, which are excluded by Magnus' theorem. A self-contained verifier is supplied as an ancillary file. The binary and unary cone tops in degree five remain open.

Author comments: 5 pages. Ancillary file anc/verify_hc4_ternary.py: 29 exact checks (python-flint, sympy, Singular via Sage), about 3 minutes. Extends arXiv:2608.14217 to degree five for ternary cone tops. Computations and writing assisted by Claude (Anthropic).