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

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

2006-09-14

arxiv.org/abs/math/0609414v3

Proc. Amer. Math. Soc. 136 (2008), No. 8, 2755-2762

Mathematics

Commutative Algebra

A few minor changes. To appear in Proc. of the AMS

Scientific paper

In this note 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 $$\lim_{r\to \infty} {f(r)\over r^{2/3}}= 6^{2/3}.$$ In general, we wonder whether our lower bound is sharp for all integers $e\geq 4$ and $r\geq 2$.

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