Classify homogeneous polynomials with identically vanishing Hessian

Determine the complete classification of homogeneous polynomials f ∈ C[x_0, …, x_{d−1}] for which the Hessian polynomial (f)—the determinant of the Hessian matrix of f—vanishes identically. This includes characterizing all dimensions d and degrees n for which such polynomials occur, extending known results (e.g., Gordan–Noether’s cones for d ≤ 4 and concise counterexamples such as Perazzo’s for d ≥ 5) to a full description of the locus.

Background

The paper studies persistent symmetric tensors (equivalently, homogeneous polynomials) and their Hessians. The Hessian (f) is the determinant of the Hessian matrix of f and is a GL(d)-equivariant invariant. Understanding when (f) vanishes identically is a classical problem connected to the geometry of the polar map and to the structure of the polynomial’s derivatives.

Classically, Gordan and Noether proved that, for d ≤ 4, hypersurfaces with vanishing Hessian are cones (non-concise polynomials). For d ≥ 5, concise polynomials with vanishing Hessian exist (e.g., Perazzo’s example). Despite these partial results, a comprehensive classification of all polynomials with vanishing Hessian across dimensions and degrees remains open.