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

Migliore, Juan; Nagel, Uwe; Zanello, Fabrizio

doi:10.1090/S0002-9939-08-09456-2

On the degree two entry of a Gorenstein $h$-vector and a conjecture of Stanley
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by Juan Migliore, Uwe Nagel and Fabrizio Zanello PDF

Proc. Amer. Math. Soc. 136 (2008), 2755-2762 Request permission

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 \[ \lim _{r\rightarrow \infty } \frac {f(r)}{ r^{2/3}}= 6^{2/3}.\] In general, we wonder whether our lower bound is sharp for all integers $e\geq 4$ and $r\geq 2$.

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