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On the degree two entry of a Gorenstein $ h$-vector and a conjecture of Stanley

Author(s): Juan Migliore; Uwe Nagel; Fabrizio Zanello
Journal: Proc. Amer. Math. Soc. 136 (2008), 2755-2762.
MSC (2000): Primary 13E10; Secondary 13H10, 13D40
Posted: April 10, 2008
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Abstract: In this short paper we establish a (non-trivial) lower bound on the degree two entry $ h_2$ of a Gorenstein $ h$-vector of any given socle degree $ e$ and any codimension $ r$.

In particular, when $ e=4$, that is, for Gorenstein $ h$-vectors of the form $ h=(1,r,h_2,r,1)$, our lower bound allows us to prove a conjecture of Stanley on the order of magnitude of the minimum value, say $ f(r)$, that $ h_2$ may assume. In fact, we show that

$\displaystyle \lim_{r\rightarrow \infty} \frac{f(r)}{ r^{2/3}}= 6^{2/3}.$

In general, we wonder whether our lower bound is sharp for all integers $ e\geq 4$ and $ r\geq 2$.

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Additional Information:

Juan Migliore
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: Juan.C.Migliore.1@nd.edu

Uwe Nagel
Affiliation: Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, Kentucky 40506-0027
Email: uwenagel@ms.uky.edu

Fabrizio Zanello
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Address at time of publication: Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan 49931-1295
Email: zanello@math.kth.se

DOI: 10.1090/S0002-9939-08-09456-2
PII: S 0002-9939(08)09456-2
Keywords: Artinian algebra, Gorenstein $h$-vector, unimodality, Green's theorem.
Received by editor(s): May 7, 2007,
Received by editor(s) in revised form: December 1, 2007
Posted: April 10, 2008
Additional Notes: The second author gratefully acknowledges partial support from and the hospitality of the Institute for Mathematics and its Applications at the University of Minnesota
Communicated by: Bernd Ulrich
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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