## 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 ≥ e_{0} = e_{0} (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 language | English |
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Pages (from-to) | 1510-1521 |

Number of pages | 12 |

Journal | Journal of Algebra |

Volume | 321 |

Issue number | 5 |

DOIs | |

State | Published - 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