Minimal resolutions of lattice ideals and integer linear programming
type: Item
instance of: scholarly article
title: Minimal resolutions of lattice ideals and integer linear programming
zbMATH Open document ID: 1094.13017
DOI: 10.4171/RMI/347
author: E. Briales-Morales
author: P. Pisón
author: A. Vigneron-Tenorio
author: Antonio Campillo
published in: Revista Matemática Iberoamericana
publication date: 2004-10-25T00:00:00Z
full work available at URL: https://eudml.org/doc/39592
review text: Let \({\mathcal L} \subseteq {\mathbb Z}^n\) be a lattice and \(I_{\mathcal L}\) be the lattice ideal associated with \({\mathcal L}\), i.e. the ideal of the polynomial ring \(k[x] = k[x_1, \dots, x_n]\) generated by the binomials \(x^{u^+}-x^{u^-}, \;u \in {\mathcal L}\) (where \(u = {u^+} - {u^-}\), \({u^+}, {u^-} \in {\mathbb N}^n\) with disjoint support). The authors show in which way the construction of the minimal free resolution of \(k[x_1, \dots, x_n]/I_{\mathcal L}\) is connected with integer linear programming. The key point is the computation of the non null reduced homology spaces of some simplicial complexes. Most of the results of the paper are obtained collecting the results already present in the vast literature on the subject; in this sense the paper is mostly a survey article.
reviewed by: Alessandro Logar
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 2109595
zbMATH Keywords: resolutions
zbMATH Keywords: simplicial complex
zbMATH Keywords: syzygy
zbMATH Keywords: lattice ideal
zbMATH Keywords: integer linear programming
describes a project that uses: GRIN
MaRDI profile type: Publication
cites work: Free resolutions and Koszul homology
cites work: Syzygies of unimodular Lawrence ideals
cites work: A criterion for detecting m-regularity
cites work: Cellular resolutions of monomial modules
cites work: Q3971568
cites work: Combinatoric of syzygies for semigroup algebras
cites work: Minimal systems of generators for ideals of semigroups
cites work: On the equations defining toric projective varieties.
cites work: Q4329403
cites work: The regularity of a toric variety
cites work: Syzygies of affine toric varieties
cites work: Higher order relations for a numerical semigroup
cites work: Q4289874
cites work: Q3136242
cites work: Toric mathematics from semigroup viewpoint.
cites work: Efficient solution of linear diophantine equations
cites work: Q4038746
cites work: An efficient incremental algorithm for solving systems of linear diophantine equations
cites work: Recent developments in toric geometry
cites work: An Algorithm to Calculate the Kernel of Certain Polynomial Ring Homomorphisms
cites work: Linear free resolutions and minimal multiplicity
cites work: Binomial ideals
cites work: On the foundations of linear and integer linear programming I
cites work: Generators and relations of abelian semigroups and semigroup rings
cites work: Q4120658
cites work: The Value-Semigroup of a One-Dimensional Gorenstein Ring
cites work: Strategies for computing minimal free resolutions
cites work: Cellular binomial ideals. Primary decomposition of binomial ideals
cites work: Index of nilpotency of binomial ideals.
cites work: On the complexity of integer programming
cites work: Generic lattice ideals
cites work: Syzygies of codimension 2 lattice ideals
cites work: The short resolution of a lattice ideal
cites work: FIRST SYZYGIES OF TORIC VARIETIES AND DIOPHANTINE EQUATIONS IN CONGRUENCE
cites work: \(\mathbb N\)-solutions to linear systems over \(\mathbb Z\)
cites work: Combinatorics and commutative algebra.
cites work: Equations Defining Toric Varieties
cites work: Q4861423
cites work: Gröbner bases of toric varieties
cites work: Semigroup ideals and linear Diophantine equations
Recommended article: Q4954923
Recommended article: Minimal resolutions of some monomial ideals
Recommended article: Linear resolutions and polymatroidal ideals
Recommended article: Minimal free resolutions of linear edge ideals
Recommended article: Q5292086
Recommended article: Minimal free resolutions of lattice ideals of digraphs
Recommended article: Minimal resolutions of algebras
Recommended article: On the Computation of Minimal Free Resolutions with Integer Coefficients
Recommended article: Laplacian ideals, arrangements, and resolutions
Recommended article: Minimal resolutions of dominant and semidominant ideals
description: scientific article; zbMATH DE number 2109595