Hilbert-series product formula for Artin–Schelter regular algebras

Determine whether the Hilbert series of every Artin–Schelter regular algebra has the product form \(H_A(t)=\prod_{i=1}^{m}(1-t^{d_i})^{-1}\) for positive integers \(d_i\).

Background

The paper studies $N$-Koszul graded algebras of finite global dimension under Kabbaj’s hypothesis that the algebra has a finite free resolution and that its Hilbert series is a product of reciprocal factors $(1-t^{d_i})$. The main theorem proves that, under this hypothesis and for $N>2$, every such algebra is 3-Koszul of global dimension 3.

A complete nonexistence result for higher-dimensional $N$-Koszul Artin–Schelter regular algebras would require knowing that every Artin–Schelter regular algebra satisfies the stated product formula for its Hilbert series. The paper identifies this as an open question attributed to Rogalski.