Introduction to Binomial Ideals

Herzog, Jürgen; Hibi, Takayuki; Ohsugi, Hidefumi

doi:10.1007/978-3-319-95349-6_3

Cited by 4 publications

(6 citation statements)

References 11 publications

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“…By [10] [Corollary 2.17], the graded Betti number β p,p+reg S/I (S/I) is always nonzero. We say that S/I is level if and only if β p (S/I) = β p,p+reg S/I (S/I).…”

Section: Cohen-macaulay Propertymentioning

confidence: 99%

Cohen–Macaulayness of Vertex Splittable Monomial Ideals

Crupi,

Ficarra

2024

Mathematics

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In this paper, we give a new criterion for the Cohen–Macaulayness of vertex splittable ideals, a family of monomial ideals recently introduced by Moradi and Khosh-Ahang. Our result relies on a Betti splitting of the ideal and provides an inductive way of checking the Cohen–Macaulay property. As a result, we obtain characterizations for Gorenstein, level and pseudo-Gorenstein vertex splittable ideals. Furthermore, we provide new and simpler combinatorial proofs of known Cohen–Macaulay criteria for several families of monomial ideals, such as (vector-spread) strongly stable ideals and (componentwise) polymatroidals. Finally, we characterize the family of bi-Cohen–Macaulay graphs by the novel criterion for the Cohen–Macaulayness of vertex splittable ideals.

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“…By [10] [Corollary 2.17], the graded Betti number β p,p+reg S/I (S/I) is always nonzero. We say that S/I is level if and only if β p (S/I) = β p,p+reg S/I (S/I).…”

Section: Cohen-macaulay Propertymentioning

confidence: 99%

Cohen–Macaulayness of Vertex Splittable Monomial Ideals

Crupi,

Ficarra

2024

Mathematics

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“…where K denotes an arbitrary field. Herzog et al in [7] and Ohtani in [8] independently introduced the notion of binomial edge ideal, and after that, the research on "binomial edge ideal" becomes a trend topic in combinatorial commutative algebra. Recently, there have been many results relating to the combinatorial data of graphs with the algebraic properties of the corresponding binomial edge ideals, see [5][6][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Regularity of Initial Ideal of Binomial Edge Ideals in Degree 2 and Their Powers

Rahim,

Saremi,

Hama

2024

Contemp. Math.

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“…An ideal I in P is called a binomial ideal if it is generated by monomials and binomials. These ideals are well-studied and occur in different contexts (see for instance [10,18]). It is therefore a natural problem to search for binomials within a given polynomial ideal.…”

Section: Introductionmentioning

confidence: 99%