Bounds and asymptotic minimal growth for Gorenstein Hilbert functions
This paper studies the problem which Hilbert functions can occur for a graded Gorenstein algebra. This problem was already solved in codimension \(r \leq 3\). In a recent paper [Proc. Am. Math. Soc. 136, No. 8, 2755--2762 (2008; Zbl 1148.13011)], the authors succeeded in computing the asymptotic growth of the least value of the Gorenstein Hilbert function in degree 2 in terms of the codimension \(r\). This result leads to two questions on the general shape of the Gorenstein Hilbert functions. The first question assumes that we know the entry of the Hilbert function in any given degree and asks for a good lower bound for the Hilbert function in the next degree. The second question ask for an asymptotic bound for the Hilbert functions given only the codimension and the socle degree. These questions are answered in this paper. From the answer to the first question one can show that given \(r\) and \(i\), all Gorenstein \(h\)-vectors of codimension \(r\) and socle degree \(e \geq e_0\) (\(e_0\) depends on \(r\) and \(i\)) are unimodal up to degree \(i+1\). In codimension 3, this result gives a new proof of a celebrated theorem of Stanley that all Gorenstein \(h\)-vectors are unimodal [\textit{P. R. Stanley}, Adv. Math. 28, 57--83 (1978; Zbl 0384.13012)]. The answer to the second question generalizes a result of Kleinschmidt on the logarithmic asymptotic behavior of the Gorenstein Hilbert function in degree \(\lfloor e/2 \rfloor\) [\textit{P. Kleinschmidt}, Arch. Math. 43, 501--506 (1984; Zbl 0533.13010)].
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