Bounds and asymptotic minimal growth for Gorenstein Hilbert functions

Juan Migliore; Uwe Nagel; Fabrizio Zanello

doi:10.1016/j.jalgebra.2008.11.026

Bounds and asymptotic minimal growth for Gorenstein Hilbert functions

Juan Migliore, Uwe Nagel, Fabrizio Zanello

Mathematics

Research output: Contribution to journal › Article › peer-review

12 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 language	English
Pages (from-to)	1510-1521
Number of pages	12
Journal	Journal of Algebra
Volume	321
Issue number	5
DOIs	https://doi.org/10.1016/j.jalgebra.2008.11.026
State	Published - Mar 1 2009

Bibliographical note

Funding Information:
E-mail addresses: [email protected] (J. Migliore), [email protected] (U. Nagel), [email protected] (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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10.1016/j.jalgebra.2008.11.026

Cite this

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title = "Bounds and asymptotic minimal growth for Gorenstein Hilbert functions",

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) ⌋.",

keywords = "Artinian algebras, Gorenstein algebras, Hilbert functions, Stanley's conjecture, Stanley's theorem, Unimodality",

author = "Juan Migliore and Uwe Nagel and Fabrizio Zanello",

note = "Funding Information: E-mail addresses: [email protected] (J. Migliore), [email protected] (U. Nagel), [email protected] (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.",

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AU - Nagel, Uwe

AU - Zanello, Fabrizio

N1 - Funding Information: E-mail addresses: [email protected] (J. Migliore), [email protected] (U. Nagel), [email protected] (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.

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N2 - 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) ⌋.

AB - 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) ⌋.

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Bounds and asymptotic minimal growth for Gorenstein Hilbert functions

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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