Gröbner Bases of Toric Ideals Associated with Matroids

In the present paper, we study the normality of the toric rings of stable set polytopes, generators of toric ideals of stable set polytopes, and their Gröbner bases via the notion of edge polytopes of finite nonsimple graphs and the results on their toric ideals. In particular, we give a criterion for the normality of the toric ring of the stable set polytope and a graph-theoretical characterization of the set of generators of the toric ideal of the stable set polytope for a graph of stability number two. As an application, we provide an infinite family of stable set polytopes whose toric ideal is generated by quadratic binomials and has no quadratic Gröbner bases.

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Toric Ideals and Their Gröbner Bases

Springer Series in Statistics - Markov Bases in Algebraic Statistics ◽

10.1007/978-1-4614-3719-2_3 ◽

2012 ◽

pp. 33-43

Author(s):

Satoshi Aoki ◽

Hisayuki Hara ◽

Akimichi Takemura

Keyword(s):

Gröbner Bases ◽

Grobner Bases ◽

Toric Ideals

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Centrally Symmetric Configurations of Integer Matrices

Nagoya Mathematical Journal ◽

10.1215/00277630-2857555 ◽

2014 ◽

Vol 216 ◽

pp. 153-170 ◽

Cited By ~ 8

Author(s):

Hidefumi Ohsugi ◽

Takayuki Hibi

Keyword(s):

Gröbner Bases ◽

Grobner Bases ◽

Toric Ideals ◽

Incidence Matrices ◽

Finite Graphs ◽

Symmetric Configuration ◽

Centrally Symmetric

AbstractThe concept of centrally symmetric configurations of integer matrices is introduced. We study the problem when the toric ring of a centrally symmetric configuration is normal and when it is Gorenstein. In addition, Gröbner bases of toric ideals of centrally symmetric configurations are discussed. Special attention is given to centrally symmetric configurations of unimodular matrices and to those of incidence matrices of finite graphs.

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Centrally Symmetric Configurations of Integer Matrices

Nagoya Mathematical Journal ◽

10.1017/s0027763000022479 ◽

2014 ◽

Vol 216 ◽

pp. 153-170 ◽

Cited By ~ 1

Author(s):

Hidefumi Ohsugi ◽

Takayuki Hibi

Keyword(s):

Gröbner Bases ◽

Grobner Bases ◽

Toric Ideals ◽

Incidence Matrices ◽

Finite Graphs ◽

Symmetric Configuration ◽

Centrally Symmetric

AbstractThe concept of centrally symmetric configurations of integer matrices is introduced. We study the problem when the toric ring of a centrally symmetric configuration is normal and when it is Gorenstein. In addition, Gröbner bases of toric ideals of centrally symmetric configurations are discussed. Special attention is given to centrally symmetric configurations of unimodular matrices and to those of incidence matrices of finite graphs.

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Gröbner Bases of Simplicial Toric Ideals

Nagoya Mathematical Journal ◽

10.1017/s002776300000979x ◽

2009 ◽

Vol 196 ◽

pp. 67-85 ◽

Cited By ~ 1

Author(s):

Michael Hellus ◽

Lê Tûan Hoa ◽

Jürgen Stückrad

Keyword(s):

Maximum Degree ◽

Gröbner Basis ◽

Gröbner Bases ◽

Lexicographic Order ◽

Grobner Bases ◽

Toric Ideals ◽

Grobner Basis

Bounds for the maximum degree of a minimal Gröbner basis of simplicial toric ideals with respect to the reverse lexicographic order are given. These bounds are close to the bound stated in Eisenbud-Goto’s Conjecture on the Castelnuovo-Mumford regularity.

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On the toric ideals of matroids of a fixed rank

Selecta Mathematica ◽

10.1007/s00029-021-00633-6 ◽

2021 ◽

Vol 27 (2) ◽

Author(s):

Michał Lasoń

Keyword(s):

Gröbner Bases ◽

Grobner Bases ◽

Toric Ideals ◽

Toric Ideal ◽

Fixed Rank

AbstractIn 1980 White conjectured that every element of the toric ideal of a matroid is generated by quadratic binomials corresponding to symmetric exchanges. We prove White’s conjecture for high degrees with respect to the rank. This extends our result (Lasoń and Michałek in Adv Math 259:1–12, 2014) confirming White’s conjecture ‘up to saturation’. Furthermore, we study degrees of Gröbner bases and Betti tables of the toric ideals of matroids of a fixed rank.

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