Solomon--Terao Algebras: Top Degree, Dimension, and Homology

We study the Solomon–Terao algebra and the Solomon–Terao complex of a central hyperplane arrangement $\mathcal{A}$ in ${\mathbb{C}}^{\mathrm{\ell }}$. Let $\eta$ be a homogeneous polynomial of degree $d\ge 2$ whose restriction to every flat of $\mathcal{A}$ has no critical point away from the origin. We prove that the Solomon–Terao algebra $\mathrm{S}\mathrm{T}(\mathcal{A},\eta )$ has top degree $\mathrm{\mid }\mathcal{A}\mathrm{\mid }+\mathrm{\ell }(d-2)$ and a one-dimensional top graded component with an explicit generator, solving a conjecture of Abe, Maeno, Murai, and Numata. For $d=2$, we show that the total dimension of $\mathrm{S}\mathrm{T}(\mathcal{A},\eta )$ is bounded below by that of the Orlik–Solomon algebra. Both results hold without assuming that the arrangement $\mathcal{A}$ is tame. Finally, we prove that $\mathcal{A}$ is tame if and only if all positive-degree homology groups of the Solomon–Terao complex associated with $\eta$ vanish.