Let $(R, \mathfrak{m})$ be a complete local ring (of dimension $2$ if that makes a difference). I would like to be able to decide whether or not $R$ is a complete intersection (meaning, a quotient of a regular local ring by a regular sequence).
The information that I allow myself is the knowledge of the Hilbert series, that is, of the function
$$n \mapsto \mathrm{length}\_R (R/\mathfrak{m}^n).$$
Is such a thing possible? More precisely, is there a criterion for $R$ to be a complete intersection in terms of the Hilbert function? More loosely, is there a sufficient (but perhaps not necessary) condition on the Hilbert function that would ensure that $R$ is a complete intersection (I am not interested in criteria that also force regularity of $R$)?


If a criterion for complete intersection is not possible, I would be willing to relax the assumptions: I would also be interested in criteria for $R$ to be Gorenstein or merely Cohen-Macaulay. Although I have found some literature on similar questions in the case when $R$ is assumed to be graded at the outset, I am not really interested in the graded case.



A criterion for complete intersection in terms of the Hilbert series?
I believe that this is an open problem and in fact one that is considered to be quite hard. See for instance [this survey](http://www.dima.unige.it/~rossim/Lectures.Rossi.Olinda.pdf) for an idea of what is known (you'll notice that even much easier question than the one you are asking are still wide open).


