Solomon–Terao Hilbert-series characterizations of free arrangements

Let ${\mathcal{A}}$ be a hyperplane arrangement, let $\eta\in U_d({\mathcal{A}})$, and write $\operatorname{Hilb}(ST({\mathcal{A}},\eta);x)$ for the Hilbert series of its Solomon–Terao algebra.

Solomon–Terao Hilbert-series conjectures. The arrangement ${\mathcal{A}}$ is free if and only if

$$\operatorname{Hilb}(ST({\mathcal{A}},\eta);x)=\prod_{i=1}^\ell(1+x+\cdots+x^{d_i})$$

for some integers $d_1,\ldots,d_\ell$; and ${\mathcal{A}}$ is free if and only if its Hilbert series is palindromic, meaning that if

$$\operatorname{Hilb}(ST({\mathcal{A}},\eta);x)=\sum_{i=0}^n a_i x^i,$$

with $a_n\neq0$, then $a_i=a_{n-i}$ for all $i$.

These conjectures seek to characterize freeness using numerical properties of Solomon–Terao algebras. The paper states that such characterizations are suggested by computations, while arrangements with factored characteristic polynomial but without freeness are already known.