Jose Martinez-Bernal

Associated primes of powers of edge ideals

Let G be a graph and let I be its edge ideal. Our main result shows that the sets of associated primes of the powers of I form an ascending chain. It is known that the sets of associated primes of Ii and

Nested Log-Concavity

{\overline{I^i}}

Minimum distance functions of graded ideals and Reed–Muller-type codes☆

stabilize for large i. We show that their corresponding stable sets are equal. To show our main result we use a classical result of Berge from matching theory and certain notions from combinatorial optimization.

TORIC IDEALS GENERATED BY CIRCUITS

Abstract Let F ={ x v 1 ,…, x v q } be a finite set of monomials in a polynomial ring R = K [ x 1 ,…, x n ] over a field K and let P be the convex hull of v 1 ,…, v q . Using linear algebra we show an expression for the relative volume of P . If v 1 ,…, v q lie in a positive hyperplane and the Rees algebra R [ Ft ] is normal, we prove the equality K [ Ft ]= A ( P ), where A ( P ) is the Ehrhart ring of P and K [ Ft ] is the monomial subring generated by Ft . We characterize, in terms of minors, when the integral closure of K [ Ft ] is equal to A ( P ).

Ehrhart Clutters: Regularity and Max-Flow Min-Cut

We give conditions on the coefficients of a polynomial p(x) so that p(x + t) be log-concave or strictly log-concave. Several applications are given: if p(x) is a polynomial with nonnegative and nondecreasing coefficients, then p(x + t) is strictly log-concave for all t ≥ 1; for any polynomial p(x) with positive leading coefficient, there is t 0 ≥ 0 such that for any t ≥ t 0 it holds that the coefficients of p(x + t) are positive, strictly decreasing, and strictly log-concave; if p(x) is a log-concave polynomial with nonnegative coefficients and no internal zeros, then p(x + t) is strictly log-concave for all t > 0; Betti numbers of lexsegment monomial ideals are strictly log-concave.

Depth and regularity of monomial ideals via polarization and combinatorial optimization

It is well known that the Stanley–Reisner ring of a matroid complex is level; this is an algebraic property between the Cohen–Macaulay and Gorenstein properties. A similar result for broken-circuit complexes is no longer true, even for graphs. We show that the Stanley–Reisner ring of the broken-circuit complex, of the cone of any simple graph, is level.