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We begin the study of the notion of diameter of an ideal I of a polynomial ring S over a field, an invariant measuring the distance between the minimal primes of I. We provide large classes of Hirsch ideals, i.e. ideals with diameter not larger than the codimension, such as: quadratic radical ideals of codimension at most 4 and such that S/I is Gorenstein, or ideals admitting a square-free complete intersection initial ideal.
We study the equality of the extremal Betti numbers of the binomial edge ideal $J_G$ and those of its initial ideal ${rm in}(J_G)$ of a closed graph $G$. We prove that in some cases there is an unique extremal Betti number for ${rm in}(J_G)$ and as a
We study the number of generators of ideals in regular rings and ask the question whether $mu(I)<mu(I^2)$ if $I$ is not a principal ideal, where $mu(J)$ denotes the number of generators of an ideal $J$. We provide lower bounds for the number of gener
Independent sets play a key role into the study of graphs and important problems arising in graph theory reduce to them. We define the monomial ideal of independent sets associated to a finite simple graph and describe its homological and algebraic i
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 different minimal sys
In this paper we develop a new technique to compute the Betti table of a monomial ideal. We present a prototype implementation of the resulting algorithm and we perform numerical experiments suggesting a very promising efficiency. On the way of descr