Odd Cycles and Hilbert Functions of Their Toric Rings

Hibi, Takayuki; Tsuchiya, Akiyoshi

doi:10.3390/math8010022

Open AccessArticle

Odd Cycles and Hilbert Functions of Their Toric Rings

by

Takayuki Hibi

¹ and

Akiyoshi Tsuchiya

^2,*

¹

Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan

²

Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8914, Japan

^*

Author to whom correspondence should be addressed.

Mathematics 2020, 8(1), 22; https://doi.org/10.3390/math8010022

Submission received: 3 December 2019 / Revised: 17 December 2019 / Accepted: 18 December 2019 / Published: 20 December 2019

(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)

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Abstract

Studying Hilbert functions of concrete examples of normal toric rings, it is demonstrated that for each

1 \leq s \leq 5

, an O-sequence

(h_{0}, h_{1}, \dots, h_{2 s - 1}) \in Z_{\geq 0}^{2 s}

satisfying the properties that (i)

h_{0} \leq h_{1} \leq \dots \leq h_{s - 1}

, (ii)

h_{2 s - 1} = h_{0}

,

h_{2 s - 2} = h_{1}

and (iii)

h_{2 s - 1 - i} = h_{i} + {(- 1)}^{i}

,

2 \leq i \leq s - 1

, can be the h-vector of a Cohen-Macaulay standard G-domain.

Keywords: O-sequence; h-vector; flawless; toric ring; stable set polytope

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MDPI and ACS Style

Hibi, T.; Tsuchiya, A. Odd Cycles and Hilbert Functions of Their Toric Rings. Mathematics 2020, 8, 22. https://doi.org/10.3390/math8010022

AMA Style

Hibi T, Tsuchiya A. Odd Cycles and Hilbert Functions of Their Toric Rings. Mathematics. 2020; 8(1):22. https://doi.org/10.3390/math8010022

Chicago/Turabian Style

Hibi, Takayuki, and Akiyoshi Tsuchiya. 2020. "Odd Cycles and Hilbert Functions of Their Toric Rings" Mathematics 8, no. 1: 22. https://doi.org/10.3390/math8010022

APA Style

Hibi, T., & Tsuchiya, A. (2020). Odd Cycles and Hilbert Functions of Their Toric Rings. Mathematics, 8(1), 22. https://doi.org/10.3390/math8010022

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Odd Cycles and Hilbert Functions of Their Toric Rings

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