Consider a polynomial ring $\mathbb C[x_1,\ldots,x_n]$ that is $\mathbb Z_{\ge 0}$-graded by degree. Let $I$ and $J$ be two homogeneous ideals therein with the same Hilbert series, i.e. with their homogeneous components of degree $n$ having the same dimension for any $n\in\mathbb Z_{\ge 0}$.
In its most general form my question is whether some kind of relationship between $I$ and $J$ follows from this condition. Or, in slightly different terms, whether one may somehow describe the set of all ideals with a given Hilbert series.
For instance, the aforementioned set of ideals is a poset with respect to taking initial ideals. Are there any general results concerning this poset?
