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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.

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I believe that this is an open problem and in fact one that is considered to be quite hard. See for instance this survey 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).

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