Author
Rafael H. Villarreal
Other affiliations: Rutgers University, Instituto Politécnico Nacional
Bio: Rafael H. Villarreal is an academic researcher from CINVESTAV. The author has contributed to research in topics: Monomial & Bipartite graph. The author has an hindex of 32, co-authored 174 publications receiving 3653 citations. Previous affiliations of Rafael H. Villarreal include Rutgers University & Instituto Politécnico Nacional.
Papers published on a yearly basis
Papers
More filters
TL;DR: In this paper, the authors studied the Cohen-Macaulay behavior of the Koszul homology of the quotient ring R/I(G) of a polynomial ring R. The main results are assertions about the Cohen -Macauley behaviour of I(G), and how normality or Cohen-macaulayness of one of the algebras can be read off the properties of the graph or in the other algebra.
365 citations
TL;DR: For a graph G and its associated ideal I(G) as discussed by the authors, a formula for the Krull dimension of the symmetric algebra of G is given along with a description of when this algebra is a domain.
Abstract: For a graph G we consider its associated ideal I(G). We uncover large classes of Cohen-Macaulay (=CM) graphs, in particular the full subclass of CM trees is presented. A formula for the Krull dimension of the symmetric algebra of I(G) is given along with a description of when this algebra is a domain. The first Koszul homology module of a CM tree is also studied.
350 citations
TL;DR: In this paper, the authors express the presentation ideal of R(I), the Rees algebra of I, in terms of the syzygies of I and its edge ideal.
Abstract: Let G be a graph and let I be its edge ideal. We express the presentation ideal of R(I), the Rees algebra of I, in terms of the syzygies of I and the presentation ideal of the special fiber of R(I). A description of the elementary integral vectors of the kernel of the incidence matrix of G is given and then used to study the special fiber of R(I) via Grobner bases.
201 citations
TL;DR: A survey of algebraic and combinatorial properties of R/I(C) and C, respectively, can be found in this paper, where the authors give a criterion to estimate the regularity of the ideal of vertex covers of C. They also examine the associated primes of powers of edge ideals and show that these sets form an ascending chain.
Abstract: Let C be a clutter and let I(C) be its edge ideal. This is a survey paper on the algebraic and combinatorial properties of R/I(C) and C, respectively. We give a criterion to estimate the regularity of R/I(C) and apply this criterion to give new proofs of some formulas for the regularity. If C is a clutter and R/I(C) is sequentially Cohen-Macaulay, we present a formula for the regularity of the ideal of vertex covers of C and give a formula for the projective dimension of R/I(C). We also examine the associated primes of powers of edge ideals, and show that for a graph with a leaf, these sets form an ascending chain.
170 citations
01 Jan 2001
TL;DR: In this article, the authors discuss the monomial algebras and its connections to combinatorics, graph theory, and polyhedral geometry, and illustrate the interplay between several areas of mathematics and the power of combinatorial commutative algebra techniques.
Abstract: Publisher Summary This chapter discusses the monomial algebras and its connections to combinatorics, graph theory, and polyhedral geometry. Some important notions from commutative algebra that have played a role in the development of the theory, such as Cohen-Macaulay ring, normal ring, Gorenstein ring, integral closure, Hilbert series, and local cohomology are introduced. The upper bound theorem for the number of faces of a simplicial sphere, a description of the integral closure of an edge subring, a generalized marriage theorem for a certain family of graphs, and a study of systems of binomials in the ideal of an affine toric variety are provided as applications. It illustrates the interplay between several areas of mathematics and the power of combinatorial commutative algebra techniques. There is a connection between monomial rings and monomial subrings due to the fact that the initial ideal of a toric ideal is a monomial ideal. This allows computing several invariants of projective varieties using algebraic systems such as CoCoA and Macaulay2. An important tool to study monomial subrings is Normaliz, which is effective in practice and can be used to find normalizations, Hilbert series, Ehrhart rings, and volumes of lattice polytopes.
116 citations
Cited by
More filters
Book•
01 Jan 2002
TL;DR: In this paper, the value of the variable in each equation is determined by a linear combination of the values of the variables in the equation and the variable's value in the solution.
Abstract: Determine the value of the variable in each equation.
635 citations