Bounds and asymptotic minimal growth for Gorenstein Hilbert functions

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We determine new bounds on the entries of Gorenstein Hilbert functions, both in any fixed codimension and asymptotically.

Our first main theorem is a lower bound for the degree i+1 entry of a Gorenstein h-vector, in terms of its entry in degree i. This result carries interesting applications concerning unimodality: indeed, an important consequence is that, given r and i, all Gorenstein h-vectors of codimension r and socle degree e⩾e0=e0(r,i) (this function being explicitly computed) are unimodal up to degree i+1. This immediately gives a new proof of a theorem of Stanley that all Gorenstein h-vectors in codimension three are unimodal.

Our second main theorem is an asymptotic formula for the least value that the ith entry of a Gorenstein h-vector may assume, in terms of codimension, r, and socle degree, e. This theorem broadly generalizes a recent result of ours, where we proved a conjecture of Stanley predicting that asymptotic value in the specific case e=4and i=2, as well as a result of Kleinschmidt which concerned the logarithmic asymptotic behavior in degree i=⌊e2⌋.

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© 2008 Elsevier Inc. All rights reserved. Publisher’s version of record: https://doi.org/10.1016/j.jalgebra.2008.11.026

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Journal of Algebra