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Normality Criteria for Monomial Ideals

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Abstract

In this paper we study the normality of monomial ideals using linear programming and graph theory. We give normality criteria for monomial ideals, for ideals generated by monomials of degree two, and for edge ideals of graphs and clutters and their ideals of covers.

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Acknowledgements

We thank the referee for a careful reading of the paper. We used Normaliz [4] and Macaulay2 [18] to give a normality test for monomial ideals and to compute Hilbert bases.

Funding

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

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Authors and Affiliations

  1. Facultad de Matemáticas, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N Zona Universitaria, Xalapa, Veracruz, 91090, Mexico

    Luis A. Dupont

  2. Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN, Apartado Postal 14–740, Mexico City, 07000, Mexico

    Humberto Muñoz-George & Rafael H. Villarreal

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  1. Luis A. Dupont
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  2. Humberto Muñoz-George
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  3. Rafael H. Villarreal
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Correspondence to Rafael H. Villarreal.

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Luis A. Dupont and Rafael H. Villarreal were supported by SNI, México. Humberto Muñoz-George was supported by a scholarship from CONACYT, México.

Appendix A: Procedures

Appendix A: Procedures

In this appendix we give procedures for Normaliz [4] and Macaulay2 [18] to determine the normality of a monomial ideal, the minimal generators of the ideal of covers of a clutter, the Hilbert basis of the Rees cone \({\mathbb {R}}_+\mathcal {A}'\) defined in Eq. (1.4), and the Hilbert basis of the cone \({\mathbb {R}}_+{\mathcal {B}}\) generated by the set \({\mathcal {B}}\) defined in Eq. (1.7). The sets \(\mathcal {A}'\) and \({\mathcal {B}}\) are used to characterize the normality of a monomial ideal (Lemma 2.2) and the normality of an ideal generated by monomials of degree two (Theorem 4.1).

Procedure A.1

Let \(I=(t^{v_1},\ldots ,t^{v_q})\) be a monomial ideal of S. The following procedure for Normaliz [4] computes the Hilbert basis of the cone generated by

$$\begin{aligned} {\mathcal {B}}=\{e_{s+1}\}\cup \{e_i+e_{s+1}\}_{i=1}^s\cup \{(v_i,1)\}_{i=1}^q, \end{aligned}$$

and determines whether or not \({\mathcal {B}}\) is a Hilbert basis. The input is the matrix whose rows are the vectors in the set \({\mathcal {B}}\). This procedure corresponds to Example 6.1.

figure a

Procedure A.2

Let \(I=(t^{v_1},\ldots ,t^{v_q})\) be a monomial ideal of S. The following procedure for Normaliz [4] computes the Hilbert basis of the Rees cone of I defined in Eq. (1.4) and determines whether or not the set \(\mathcal {A}'=\{e_i\}_{i=1}^s\cup \{(v_i,1)\}_{i=1}^q\) is a Hilbert basis. In particular, by Lemma 2.2, we can determine whether or not I is a normal ideal. The input is the matrix with rows \(v_1,\ldots ,v_q\). This procedure corresponds to Example 6.1.

figure b

Procedure A.3

(Normality test). Let I be a monomial ideal. We implement a procedure—that uses the interface of Macaulay2 [18] to Normaliz [4]—to determine the normality of I, and the normality and minimal generators of the ideal of covers of a clutter. This procedure corresponds to Example 6.2. To compute other examples, in the next procedure simply change the polynomial rings R and S, and the generators of I.

figure c

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Dupont, L.A., Muñoz-George, H. & Villarreal, R.H. Normality Criteria for Monomial Ideals. Results Math 78, 34 (2023). https://doi.org/10.1007/s00025-022-01814-1

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