Combinatorics and Computation in Commutative Algebra

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: closed (29 February 2024) | Viewed by 12000

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Mathematics Research Institute (IMUVA), University of Valladolid, Facultad de Ciencias, Paseo Belén, 7, 47011 Valladolid, Spain
Interests: commutative algebra; computation and combinatorics in commutative algebra and algebraic geometry; syzygies; semigroup rings; monomial and toric ideals; edge ideal; commutative algebra and coding theory

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Departamento de Matemáticas, Estadística e I.O. Universidad de La Laguna. C/ Astrofísico Francisco Sánchez s/n, 38200 La Laguna, Spain
Interests: commutative algebra; semigroup rings; toric ideals; combinatorial commutative algebra; Gröbner bases; Cayley graphs; coding theory

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Departamento de Matemáticas y Computación, Universidad de La Rioja, La Rioja, Spain
Interests: commutative algebra; computation and combinatorics in commutative algebra and algebraic geometry; reliability theory; monomial ideals; edge ideal; networks; computer science

Special Issue Information

Commutative algebra is a classical area of mathematics that studies algebraic structures over commutative rings. Following the fundamental works of R. Dedekind, D. Hilbert, E. Noether and W. Krull, among others, it became an independent field in the 1930s. One of the most outstanding starting points was the work of Hilbert on ideals in a polynomial ring and their free resolutions, a topic that has been a permanently active line of research ever since.

From its early stage, commutative algebra has also had deep interactions with other disciplines of mathematics such as algebraic geometry, number theory, representation theory, algebraic topology and, more recently, algebraic combinatorics, computational algebra, coding theory or cryptography.

Commutative algebra has, in particular, a strong interplay with combinatorics, from which it extracts and to which it transfers ideas, results, and techniques. On the other hand, algorithmic methods have acquired an important role in commutative algebra due to the development of techniques based on Gröbner bases, which have allowed the creation of powerful algorithms.

The aim of this Special Issue of Mathematics is to show recent trends on combinatorial and computational aspects of commutative algebra and its applications. We cordially invite you to present your recent contributions to this Special Issue.

Prof. Dr. Philippe Gimenez
Prof. Dr. Ignacio García Marco
Prof. Dr. Eduardo Sáenz De Cabezón
Guest Editors

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Keywords

  • free resolutions, syzygies, Betti numbers
  • castelnuovo–mumford regularity
  • monomial and toric ideals
  • ideals and algebras associated to graphs
  • simplicial complexes in commutative algebra
  • algorithms in commutative algebra and their implementation
  • matroids, posets, polytopes and codes in commutative algebra
  • applications

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Published Papers (6 papers)

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Research

14 pages, 297 KiB  
Article
Cohen–Macaulayness of Vertex Splittable Monomial Ideals
by Marilena Crupi and Antonino Ficarra
Mathematics 2024, 12(6), 912; https://doi.org/10.3390/math12060912 - 20 Mar 2024
Viewed by 857
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Combinatorics and Computation in Commutative Algebra)
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17 pages, 372 KiB  
Article
Depth and Stanley Depth of the Edge Ideals of r-Fold Bristled Graphs of Some Graphs
by Ying Wang, Sidra Sharif, Muhammad Ishaq, Fairouz Tchier, Ferdous M. Tawfiq and Adnan Aslam
Mathematics 2023, 11(22), 4646; https://doi.org/10.3390/math11224646 - 14 Nov 2023
Viewed by 844
Abstract
In this paper, we find values of depth, Stanley depth, and projective dimension of the quotient rings of the edge ideals associated with r-fold bristled graphs of ladder graphs, circular ladder graphs, some king’s graphs, and circular king’s graphs. Full article
(This article belongs to the Special Issue Combinatorics and Computation in Commutative Algebra)
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15 pages, 325 KiB  
Article
Free Resolutions and Generalized Hamming Weights of Binary Linear Codes
by Ignacio García-Marco, Irene Márquez-Corbella, Edgar Martínez-Moro and Yuriko Pitones
Mathematics 2022, 10(12), 2079; https://doi.org/10.3390/math10122079 - 15 Jun 2022
Cited by 1 | Viewed by 2451
Abstract
In this work, we explore the relationship between the graded free resolution of some monomial ideals and the Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure that is smaller than the set of codewords of minimal support [...] Read more.
In this work, we explore the relationship between the graded free resolution of some monomial ideals and the Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure that is smaller than the set of codewords of minimal support that provides us with some information about the GHWs. We prove that the first and second generalized Hamming weights of a binary linear code can be computed (by means of a graded free resolution) from a set of monomials associated with a binomial ideal related with the code. Moreover, the remaining weights are bounded above by the degrees of the syzygies in the resolution. Full article
(This article belongs to the Special Issue Combinatorics and Computation in Commutative Algebra)
11 pages, 284 KiB  
Article
Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves
by Manuel B. Branco, Isabel Colaço and Ignacio Ojeda
Mathematics 2021, 9(24), 3204; https://doi.org/10.3390/math9243204 - 11 Dec 2021
Cited by 4 | Viewed by 2000
Abstract
Let a,b and n>1 be three positive integers such that a and j=0n1bj are relatively prime. In this paper, we prove that the toric ideal I associated to the submonoid of [...] Read more.
Let a,b and n>1 be three positive integers such that a and j=0n1bj are relatively prime. In this paper, we prove that the toric ideal I associated to the submonoid of N generated by {j=0n1bj}{j=0n1bj+aj=0i2bji=2,,n} is determinantal. Moreover, we prove that for n>3, the ideal I has a unique minimal system of generators if and only if a<b1. Full article
(This article belongs to the Special Issue Combinatorics and Computation in Commutative Algebra)
13 pages, 302 KiB  
Article
Polynomial and Pseudopolynomial Procedures for Solving Interval Two-Sided (Max, Plus)-Linear Systems
by Helena Myšková and Ján Plavka
Mathematics 2021, 9(22), 2951; https://doi.org/10.3390/math9222951 - 18 Nov 2021
Cited by 2 | Viewed by 1720
Abstract
Max-plus algebra is the similarity of the classical linear algebra with two binary operations, maximum and addition. The notation Ax = Bx, where A, B are given (interval) matrices, represents (interval) two-sided (max, plus)-linear system. For the solvability of Ax = Bx, there [...] Read more.
Max-plus algebra is the similarity of the classical linear algebra with two binary operations, maximum and addition. The notation Ax = Bx, where A, B are given (interval) matrices, represents (interval) two-sided (max, plus)-linear system. For the solvability of Ax = Bx, there are some pseudopolynomial algorithms, but a polynomial algorithm is still waiting for an appearance. The paper deals with the analysis of solvability of two-sided (max, plus)-linear equations with inexact (interval) data. The purpose of the paper is to get efficient necessary and sufficient conditions for solvability of the interval systems using the property of the solution set of the non-interval system Ax = Bx. The main contribution of the paper is a transformation of weak versions of solvability to either subeigenvector problems or to non-interval two-sided (max, plus)-linear systems and obtaining the equivalent polynomially checked conditions for the strong versions of solvability. Full article
(This article belongs to the Special Issue Combinatorics and Computation in Commutative Algebra)
16 pages, 389 KiB  
Article
Induced Matchings and the v-Number of Graded Ideals
by Gonzalo Grisalde, Enrique Reyes and Rafael H. Villarreal
Mathematics 2021, 9(22), 2860; https://doi.org/10.3390/math9222860 - 11 Nov 2021
Cited by 10 | Viewed by 1859
Abstract
We give a formula for the v-number of a graded ideal that can be used to compute this number. Then, we show that for the edge ideal I(G) of a graph G, the induced matching number of G is [...] Read more.
We give a formula for the v-number of a graded ideal that can be used to compute this number. Then, we show that for the edge ideal I(G) of a graph G, the induced matching number of G is an upper bound for the v-number of I(G) when G is very well-covered, or G has a simplicial partition, or G is well-covered connected and contains neither four, nor five cycles. In all these cases, the v-number of I(G) is a lower bound for the regularity of the edge ring of G. We classify when the induced matching number of G is an upper bound for the v-number of I(G) when G is a cycle and classify when all vertices of a graph are shedding vertices to gain insight into the family of W2-graphs. Full article
(This article belongs to the Special Issue Combinatorics and Computation in Commutative Algebra)
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