Odd Cycles and Hilbert Functions of Their Toric Rings

Studying Hilbert functions of concrete examples of normal toric rings, it is demonstrated that for each 1 ≤ s ≤ 5 , an O-sequence ( h 0 , h 1 , … , h 2 s − 1 ) ∈ Z ≥ 0 2 s satisfying the properties that (i) h 0 ≤ h 1 ≤ ⋯ ≤ 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 ≤ i ≤ s − 1 , can be the h-vector of a Cohen-Macaulay standard G-domain.

Toric Rings and Ideals of Stable Set Polytopes

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.

Non-Koszul quadratic Gorenstein toric rings

Koszulness of Gorenstein quadratic algebras of small socle degree is studied. In this paper, we construct non-Koszul Gorenstein quadratic toric ring such that its socle degree is more than $3$ by using stable set polytopes.

Reverse Lexicographic Gröbner Bases and Strongly Koszul Toric Rings

Restuccia and Rinaldo proved that a standard graded $K$-algebra $K[x_1,\dots,x_n]/I$ is strongly Koszul if the reduced Gröbner basis of $I$ with respect to any reverse lexicographic order is quadratic. In this paper, we give a sufficient condition for a toric ring $K[A]$ to be strongly Koszul in terms of the reverse lexicographic Gröbner bases of its toric ideal $I_A$. This is a partial extension of a result given by Restuccia and Rinaldo. In addition, we show that any strongly Koszul toric ring generated by squarefree monomials is compressed. Using this fact, we show that our sufficient condition for $K[A]$ to be strongly Koszul is both necessary and sufficient when $K[A]$ is generated by squarefree monomials.