Related papers: Arithmetical rank of toric ideals associated to gr…

To any lattice $L \subset \mathbb{Z}^{m}$ one can associate the lattice ideal $I_{L} \subset K[x_{1},...,x_{m}]$. This paper concerns the study of the relation between the binomial arithmetical rank and the minimal number of generators of…

Let $G$ be a simple graph on the vertex set $\{1,\ldots,n\}$ with $m$ edges. An algebraic object attached to $G$ is the ideal $P_{G}$ generated by diagonal 2-minors of an $n \times n$ matrix of variables. In this paper we prove that if $G$…

Let $G$ be a finite connected simple graph and $I_{G}$ the toric ideal of the edge ring $K[G]$ of $G$. In the present paper, we study finite graphs $G$ with the property that $I_{G}$ is generated by quadratic binomials and $I_{G}$ possesses…

Let $G$ be a finite simple graph and let $I_G$ denote its associated toric ideal in the polynomial ring $R$. For each integer $n\geq 2$, we completely determine all the possible values for the tuple $({\rm reg}(R/I_G), {\rm…

In the present paper, we investigate the maximal degree of minimal generators of the toric ideal of the matching polytope of a graph. It is known that the toric ideal associated to a bipartite graph is generated by binomials of degree at…

A graph-theoretic method, simpler than existing ones, is used to characterize the minimal set of monomial generators for the integral closure of any algebra of polynomials generated by quadratic monomials. The toric ideal of relations…

Let $I_G$ be the toric ideal of a graph $G$. We characterize in graph theoretical terms the primitive, the minimal, the indispensable and the fundamental binomials of the toric ideal $I_G$.

Let $G$ be a bipartite graph and $I(G)$ the toric ideal associated to the graph $G$. In this article we calculate Hilbert-Samuel multiplicity of the graph $G$ for which the toric ideal $I(G)$ is generated by a quadratic binomials and it…

Let $G$ be a finite simple graph. We give a lower bound for the Castelnuovo-Mumford regularity of the toric ideal $I_G$ associated to $G$ in terms of the sizes and number of induced complete bipartite graphs in $G$. When $G$ is a chordal…

Let $G$ be a simple graph on the vertex set $\{v_{1},\ldots,v_{n}\}$. An algebraic object attached to $G$ is the toric ideal $I_G$. We say that $I_G$ is subgraph splittable if there exist subgraphs $G_1$ and $G_2$ of $G$ such that…

Let G be a finite simple graph. In this paper we will show that the binomial edge ideal of G, JG is toric if and only if each connected component of G is complete and in this case it is the sum of toric ideal associated to bipartite…

We study the problem when an ideal generated by adjacent 2-minors is the toric ideal of a finite graph.

In this paper, we investigate the arithmetical rank of a binomial ideal $J$. We provide lower bounds for the binomial arithmetical rank and the $J$-complete arithmetical rank of $J$. Special attention is paid to the case where $J$ is the…

Let $G$ be a connected and simple graph on the vertex set $[n]$. To the graph $G$ one can associate the generalized binomial edge ideal $J_{m}(G)$ in the polynomial ring $R=K[x_{ij}: i \in [m], j \in [n]]$. We provide a lower bound for the…

Describing families of ideals that are minimally generated by at least one, or by all, of their reduced Gr\"obner bases is a central topic in commutative algebra. In this paper, we address this problem in the context of toric ideals of…

Given a vertex-weighted oriented graph, we can associate to it a set of monomials. We consider the toric ideal whose defining map is given by these monomials. We find a generating set for the toric ideal for certain classes of graphs which…

Let X be an algebraic toric set in a projective space over a finite field. We study the vanishing ideal, I(X), of X and show some useful degree bounds for a minimal set of generators of I(X). We give an explicit description of a set of…

Let $A=\{{\bf a}_1,...,{\bf a}_m\} \subset \mathbb{Z}^n$ be a vector configuration and $I_A \subset K[x_1,...,x_m]$ its corresponding toric ideal. The paper consists of two parts. In the first part we completely determine the number of…

Let $I \subseteq R = \mathbb{K}[x_1,\ldots,x_n]$ be a toric ideal, i.e., a binomial prime ideal. We investigate when the ideal $I$ can be "split" into the sum of two smaller toric ideals. For a general toric ideal $I$, we give a sufficient…

We investigate the structure of ideals generated by binomials (polynomials with at most two terms) and the schemes and varieties associated to them. The class of binomial ideals contains many classical examples from algebraic geometry, and…