TY - JOUR
T1 - On the degree two entry of a Gorenstein h-vector and a conjecture of Stanley
AU - Migliore, Juan
AU - Nagel, Uwe
AU - Zanello, Fabrizio
PY - 2008/8
Y1 - 2008/8
N2 - In this short paper we establish a (non-trivial) lower bound on the degree two entry h2 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,h2,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 h2 may assume. In fact, we show that lim r→∞f(r)/r2/3=62/3. In general, we wonder whether our lower bound is sharp for all integers e ≥ 4 and r ≥ 2.
AB - In this short paper we establish a (non-trivial) lower bound on the degree two entry h2 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,h2,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 h2 may assume. In fact, we show that lim r→∞f(r)/r2/3=62/3. In general, we wonder whether our lower bound is sharp for all integers e ≥ 4 and r ≥ 2.
KW - Artinian algebra
KW - Gorenstein h-vector
KW - Green's theorem
KW - Unimodality
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U2 - 10.1090/S0002-9939-08-09456-2
DO - 10.1090/S0002-9939-08-09456-2
M3 - Article
AN - SCOPUS:58149505072
SN - 0002-9939
VL - 136
SP - 2755
EP - 2762
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
IS - 8
ER -