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I am reading through Richard Stanley's 1990 paper "On the Hilbert Function of a Graded Cohen-Macaulay Domain" to present in a seminar. I am trying to provide a reasonable conclusion for this talk, and would prefer to mention some open problems. But I am not very well-versed in the literature— and since the paper is relatively old, I am not sure if any of the problems are still open.

Questions:

  • In section 1, Stanley mentions that some partial characterization of Hilbert series is known for the class of standard connected Gorenstein rings. Have we done any better since then?
  • The main Theorem 2.1 gives some necessary conditions on the $h$-vector of a semistandard graded Cohen-Macaulay domain. Is this still the state of the art? How about for rings coming from polytopes as in Section 4?
  • In the discussion after Proposition 3.4, he suggests that there are no counterexamples known in positive characteristic. Has this been proven, or counterexamples found?

I've skimmed through one level mathscinet citations, to no avail.

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Perhaps this is too late for your seminar, but there has been a huge literature on these problems and related ones. The key words to search are: "h-vector of...". For example, here is a paper that deals with Gorenstein domains of codimension three. Here is a paper on h-vector of Gorenstein toric ring, which Section 4 of Stanley's paper addressed.

In general, even for Artinian Gorenstein algebras, a complete chracterization of h-vector seems to be out of reach, but there are many surveys and works are being posted at this very moment.

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