Related papers: The Circuit Ideal of a Vector Configuration

Let I be the toric ideal of a homogeneous normal configuration. We prove that I is generated by circuits if and only if each unbalanced circuit of I has a "connector" which is a linear combination of circuits with a square-free term. In…

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…

In this paper, we study toric ideals generated by circuits. For toric ideals which have squarefree quadratic initial ideals, a sufficient condition to be generated by circuits is given. In particular, squarefree Veronese subrings, the…

Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I \subset S$ a monomial ideal. Given a vector $\mathfrak{c}\in\mathbb{N}^n$, the ideal $I_{\mathfrak{c}}$ is the ideal generated by those monomials…

In this work, we classify the circuit binomials of any weighted oriented graph $D$ and we explicitly compute the circuit binomials of $D$ in terms of the minors of the incidence matrix of $D$. We show that the circuit binomials of any…

In this paper we derive some basic results of circuit theory using `Implicit Linear Algebra' (ILA). This approach has the advantage of simplicity and generality. Implicit linear algebra is outlined in [1]. We denote the space of all vectors…

We characterize the graphs $G$ for which their toric ideals $I_G$ are complete intersections. In particular we prove that for a connected graph $G$ such that $I_G$ is complete intersection all of its blocks are bipartite except of at most…

We study the set of circuits of a homogeneous ideal and that of its truncations, and introduce the notion of generic circuits set. We show how this is a well-defined invariant that can be used, in the case of initial ideals with respect to…

A proper ideal $I$ in a commutative ring with unity is called a $z^\circ$-ideal if for each $a$ in $I$, the intersection of all minimal prime ideals in $R$ which contain $a$ is contained in $I$. For any totally ordered field $F$ and a…

A normal (respectively, graded normal) vector configuration $A$ defines the toric ideal $I_A$ of a normal (respectively, projectively normal) toric variety. These ideals are Cohen-Macaulay, and when $A$ is normal and graded, $I_A$ is…

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…

To any toric ideal $I_A$, encoded by an integer matrix $A$, we associate a matroid structure called {\em the bouquet graph} of $A$ and introduce another toric ideal called {\em the bouquet ideal} of $A$. We show how these objects capture…

We study minimal reductions of edge ideals of graphs and determine restrictions on the coefficients of the generators of these minimal reductions. We prove that when $I$ is not basic, then $\core{I}\subset \m I$, where $I$ is an edge ideal…

Let $R$ be a finite commutative ring with identity, and let $P$ be a proper prime ideal of $R$. The prime ideal graph $\Gamma_P(R)$ has vertex set of $R\setminus\{0\}$, where two distinct vertices $x$ and $y$ are adjacent if and only if…

For $a\in R$, let $P_a$ denote the intersection of all minimal prime ideals of $R$ containing $a$. An ideal $I$ of a ring $R$ is called a $z^{\circ}$-ideal if $P_a\subseteq I$ for all $a\in I$. In this paper, we first investigate the class…

Completely prime right ideals are introduced as a one-sided generalization of the concept of a prime ideal in a commutative ring. Some of their basic properties are investigated, pointing out both similarities and differences between these…

Let $\Gamma$ be the unit circle, $A(\Gamma)$ the Wiener algebra of continuous functions whose series of Fourier coefficients are absolutely convergent, and $A^+$ the subalgebra of $A(\Gamma)$ of functions whose negative coefficients are…

We show that any nonzero polynomial in the ideal generated by the $r \times r$ minors of an $n \times n$ matrix $X$ can be used to efficiently approximate the determinant. For any nonzero polynomial $f$ in this ideal, we construct a small…

Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and let $I \subset S$ be a monomial ideal. For a vector $\mathfrak{c}\in\mathbb{N}^n$, we set $I_{\mathfrak{c}}$ to be the ideal generated by monomials…

For an ideal $I$ in a Noetherian ring $R$, we introduce and study its conductor as a tool to explore the Rees algebra of $I$. The conductor of $I$ is an ideal $C(I)\subset R$ obtained from the defining ideals of the Rees algebra and the…