Journal of the
Korean Mathematical Society
JKMS

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

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J. Korean Math. Soc. 2015; 52(4): 685-697

Printed July 1, 2015

https://doi.org/10.4134/JKMS.2015.52.4.685

Copyright © The Korean Mathematical Society.

On the $(n, d)^{th}$ $f$-ideals

Jin Guo and Tongsuo Wu

Hainan University, Shanghai Jiaotong University

Abstract

For a field $K$, a square-free monomial ideal $I$ of $K[x_1,\ldots,x_n]$ is called an {\it $f$-ideal}, if both its facet complex and Stanley-Reisner complex have the same $f$-vector. Furthermore, for an $f$-ideal $I$, if all monomials in the minimal generating set $G(I)$ have the same degree $d$, then $I$ is called an $(n, d)^{th}$ $f$-ideal. In this paper, we prove the existence of $(n, d)^{th}$ $f$-ideal for $d \geq 2$ and $n \geq d+2$, and we also give some algorithms to construct $(n, d)^{th}$ $f$-ideals.

Keywords: perfect set, $f$-ideal, unmixed $f$-ideal, perfect number

MSC numbers: 13P10, 13F20, 13C14, 05A18

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