Remark 2

Conversely, the fact that there exist integers $d_i > 0$ such that $P_H = \prod_i (1 - T^{d_i})^{-1}$ does not imply that $H$ is a graded polynomial algebra; for example, if $H$ is generated by an element $X$ of degree 1 and an element $Y$ of degree 2, subject only to the relation $X^2 = 0$, one has $P_H = (1 - T)^{-1}$.