Bounds and asymptotic minimal growth for Gorenstein Hilbert functions

Juan Migliore, Uwe Nagel, Fabrizio Zanello

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

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 = 4 and i = 2, as well as a result of Kleinschmidt which concerned the logarithmic asymptotic behavior in degree i = ⌊ frac(e, 2) ⌋.

Original languageEnglish
Pages (from-to)1510-1521
Number of pages12
JournalJournal of Algebra
Volume321
Issue number5
DOIs
StatePublished - Mar 1 2009

Bibliographical note

Funding Information:
E-mail addresses: juan.c.migliore.1@nd.edu (J. Migliore), uwenagel@ms.uky.edu (U. Nagel), zanello@math.kth.se (F. Zanello). 1 Part of the work for this paper was done while the author was sponsored by the National Security Agency under Grant Number H98230-07-1-0036. 2 Part of the work for this paper was done while the author was sponsored by the National Security Agency under Grant Number H98230-07-1-0065. 3 The idea of this work originated from a discussion between the second and the third author, during a workshop at the Institute for Mathematics and Applications (IMA). They both thank the IMA for partial support.

Keywords

  • Artinian algebras
  • Gorenstein algebras
  • Hilbert functions
  • Stanley's conjecture
  • Stanley's theorem
  • Unimodality

ASJC Scopus subject areas

  • Algebra and Number Theory

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