Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2022; 59(1): 71-85

Online first article January 1, 2022      Printed January 1, 2022

https://doi.org/10.4134/JKMS.j200690

Copyright © The Korean Mathematical Society.

Gorenstein sequences of high socle degrees

Jung Pil Park, Yong-Su Shin

Seoul National University; Sungshin Women's University

Abstract

In [4], the authors showed that if an $h$-vector $(h_0,h_1,\dots,h_e)$ with $h_1=4e-4$ and $h_i\le h_1$ is a Gorenstein sequence, then $h_1=h_i$ for every $1\le i\le e-1$ and $e\ge 6$. In this paper, we show that if an $h$-vector $(h_0,h_1,\dots,h_e)$ with $h_1=4e-4$, $h_2=4e-3$, and $h_i\le h_2$ is a Gorenstein sequence, then $h_2=h_i$ for every $2\le i\le e-2$ and $e\ge 7$. We also propose an open question that if  an $h$-vector $(h_0,h_1,\dots,h_e)$ with $h_1=4e-4$, $4e-3<h_2\le {(h_1)_{(1)}}| ^{+1}_{+1}$, and $h_2\le h_i$ is a Gorenstein sequence, then $h_2=h_i$ for every $2\le i\le e-2$ and $e\ge 6$.

Keywords: Gorenstein $h$-vectors, unimodal or nonunimodal $h$-vectors, Hilbert functions

MSC numbers: Primary 13D40; Secondary 13H10, 14C20

Supported by: This research was supported by a grant from Sungshin Women's University.