Alexandru Constantinescu

Hilbert Function and Betti Numbers of Algebras with Lefschetz Property of Order m

The authors in Harima et al. (2003) characterize the Hilbert function of algebras with the Lefschetz property. We extend this characterization to algebras with the Lefschetz property m times. We also give upper bounds for the Betti numbers of Artinian algebras with a given Hilbert function and with the Lefschetz property m times and describe the cases in which these bounds are reached.

h-Vectors of Matroid Complexes

In this paper we partition in classes the set of matroids of fixed dimension on a fixed vertex set. In each class we identify two special matroids, respectively with minimal and maximal h-vector in that class. Such extremal matroids also satisfy a long-standing conjecture of Stanley. As a byproduct of this theory we establish Stanley’s conjecture in various cases, for example the case of Cohen-Macaulay type less than or equal to 3.

Cohen–Macaulayness and Limit Behavior of Depth for Powers of Cover Ideals

Let 𝕂 be a field, and let R = 𝕂[x 1,…, x n ] be the polynomial ring over 𝕂 in n indeterminates x 1,…, x n . Let G be a graph with vertex-set {x 1,…, x n }, and let J be the cover ideal of G in R. For a given positive integer k, we denote the kth symbolic power and the kth bracket power of J by J (k) and J [k], respectively. In this paper, we give necessary and sufficient conditions for R/J k , R/J (k), and R/J [k] to be Cohen–Macaulay. We also study the limit behavior of the depths of these rings.

Generic and special constructions of pure O-sequences

It is shown that the h-vectors of Stanley–Reisner rings of three classes of matroids are pure O-sequences. The classes are (a) matroids that are truncations of matroids, or more generally of Cohen–Macaulay complexes, (b) matroids whose dual is (rank + 2)-partite, and (c) matroids of Cohen–Macaulay type at most 5. Consequences for the computational search for a counterexample to a conjecture of Stanley are discussed.

Veronese algebras and modules of rings with straightening laws

Do the Veronese rings of an algebra with straightening laws (ASL) still have an ASL structure? We give positive answers to this question in some particular cases, namely for the second Veronese algebra of Hibi rings and of discrete ASLs. We also prove that the Veronese modules of the polynomial ring have a structure of module with straightening laws. In dimension at most three we present a poset construction that has the required combinatorial properties to support such a structure.