arXiv subjects

Guillermo Alesandroni

Publications and source records attributed to Guillermo Alesandroni.

10 recordsLinked to original sources

arXiv2020

Families of finite sets in which no set is covered by the union of the others

Let F be a finite nonempty family of finite nonempty sets. We prove the following: (i) F satisfies the condition of the title if and only if for every pair of distinct subfamilies {A_1,...,A_r}, {B_1,...,B_s} of F, the union of the A_i is different from the union of the B_i. (ii) If F satisfies the condition of the title, then the number of subsets of the union of the members of F containing at least one set of F is odd. We give two applications of these results, one to number theory and one to commutative algebra.

math.CO↗

arXiv2020

The Erdos-Faber-Lovasz conjecture for weakly dense hypergraphs

Generalizing the concept of dense hypergraph, we say that a hypergraph is weakly dense, if no k in the half-open interval [2,sqrt(n)) is the degree of more than k^2 vertices. In our main result, we prove the famous Erdos-Faber-Lovasz conjecture when the hypergraph is weakly dense.

math.CO↗

arXiv2019

Monomial invariants applied to graph coloring

This article is built upon three main ideas. First, for a class of monomial ideals, it is proven that the multiplicity of an ideal equals the number of realizations of its codimension (an intuitive concept that we define later). Next, for an arbitrary graph G, we construct a monomial ideal M_G, and show that the chromatic number of G is equal to the codimension of M_G. Finally, for a class of graphs, we give a formula that computes the chromatic polynomial of G, evaluated at the chromatic number of G, in terms of the codimension and multiplicity of M_G. In particular, the formula applies to all graphs satisfying the Erdos-Faber-Lovász conjecture.

math.AC↗

arXiv2019

The order of dominance of a monomial ideal

Let S be a polynomial ring in n variables over a field, and let M be a monomial ideal of S. We introduce a new invariant, called the order of dominance of S/M, denoted odom(S/M), which has many similarities with the codimension of S/M. We use this order of dominance to characterize the class of Scarf ideals that are Cohen-Macaulay, and also to characterize when the Taylor resolution is minimal. We also show that odom(S/M) has the following properties: (i) codim(S/M) <= odom(S/M) <= pd(S/M). (ii) pd(S/M)=n if and only if odom(S/M)=n. (iii) pd(S/M)=1 if and only if odom(S/M)=1. (iv) If odom(S/M)=n-1 then pd(S/M)=n-1.

math.AC↗

arXiv2019

Betti numbers of monomial ideals in four variables

We express the multigraded Betti numbers of monomial ideals in 4 variables in terms of the multigraded Betti numbers of 66 squarefree monomial ideals, also in 4 variables. We use this class of 66 ideals to prove that monomial resolutions in 4 variables are independent of the base field. In addition, we give a formula for the Betti numbers of an arbitrary monomial ideal in 4 variables.

math.AC↗

arXiv2019

Monomial multiplicities in explicit form

In this article we give explicit descriptions of the multiplicities of some classes of monomial ideals. For instance, we give a formula for the multiplicities of all codimension 1 monomial ideals, and another formula for the multiplicities of almost complete intersections. We also introduce a new class of ideals that extends the family of monomial complete intersections, and give a formula for their multiplicity, as well as a visual interpretation of this invariant.

math.AC↗

arXiv2017

Hilbert's Syzygy Theorem for monomial ideals

We give a new proof of Hilbert's Syzygy Theorem for monomial ideals. In addition, we prove the following. If S=k[x_1,...,x_n] is a polynomial ring over a field, M is a squarefree monomial ideal in S, and each minimal generator of M has degree larger than i, then the projective dimension of S/M is at most n-i.

math.AC↗

arXiv2017

Monomial ideals with large projective dimension

Let S be a polynomial ring in n variables, over an arbitrary field. We give the total, graded, and multigraded Betti numbers of S/M, for every monomial ideal M in S. We also give an explicit characterization of all monomial ideals M in S for which the quotient S/M has projective dimension n.

math.AC↗

arXiv2017

Structural decomposition of monomial resolutions

We express the multigraded Betti numbers of an arbitrary monomial ideal in terms of the multigraded Betti numbers of two basic classes of ideals. This decompo- sition has multiple applications. In some concrete cases, we use it to construct minimal resolutions of classes of monomial ideals; in other cases, we use it to compute projective dimensions. To illustrate the effectiveness of the structural decomposition, we give a new proof of a classic theorem by Charalambous.

math.AC↗

arXiv2014

Minimal Resolutions of Dominant and Semidominant Ideals

We construct the minimal resolutions of three classes of monomial ideals: dominant, 1-semidominant, and 2-semidominant ideals. The families of dominant and 1-semidominant ideals extend those of complete and almost complete intersections. We show that dominant ideals give a precise characterization of when the Taylor resolution is minimal, 1-semidominant ideals are Scarf, and the minimal resolutions of 2-semidominant ideals can be obtained from their Taylor resolutions by eliminating faces and facets of equal multidegree, in arbitrary order. We study the combinatorial properties of these classes of ideals and explain how they relate to generic ideals.

math.AC↗