Title
Bounds and asymptotic minimal growth for Gorenstein Hilbert functions
Document Type
Article
Publication Date
3-2009
Abstract
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⌋.
Publication Title
Journal of Algebra
Recommended Citation
Migliore, J.,
Nagel, U.,
&
Zanello, F.
(2009).
Bounds and asymptotic minimal growth for Gorenstein Hilbert functions.
Journal of Algebra,
321(5), 1510-1521.
http://doi.org/10.1016/j.jalgebra.2008.11.026
Retrieved from: https://digitalcommons.mtu.edu/math-fp/41
Publisher's Statement
© 2008 Elsevier Inc. All rights reserved. Publisher’s version of record: https://doi.org/10.1016/j.jalgebra.2008.11.026