Marius Vladoiu

Stanley depth and size of a monomial ideal

Lyubeznik introduced the concept of size of a monomial ideal and showed that the size of a monomial ideal increased by

Ideals of fiber type and polymatroids

1

Minimal generating sets of lattice ideals

is a lower bound for its depth. We show that the size is also a lower bound for its Stanley depth. Applying Alexander duality we obtain upper bounds for the regularity and Stanley regularity of squarefree monomial ideals.

MARKOV BASES AND GENERALIZED LAWRENCE LIFTINGS

In the first half of this paper, we complement the theory on disc rete polymatroids. More precisely, (i) we prove that a discrete polymatro id satisfying the strong exchange property is, up to an affinity, of Veronese type; (ii ) we classify all uniform matroids which are level; (iii) we introduce the concept of ideals of fiber type and show that all polymatroidal ideals are of fiber type. On the oth er hand, in the latter half of this paper, we generalize the result proved by Stefan Blum that the defining ideal of the Rees ring of a base sortable matroid possesses a quadratic Gr¨ obner basis. For this purpose we introduce the concept of “ -exchange property” and show that a Gr¨ obner basis of the defining ideal of the Rees ring of an ideal ca n be determined and that is of fiber type if satisfies the -exchange pr operty. Ideals satisfying the -exchange property include strongly stable ideals, polymatroid ideals of base sortable discrete polymatroids, ideals of Segre-Veronese type and certain ideals related to classical root systems.

How to compute the Stanley depth of a monomial ideal

Let

The stable set of associated prime ideals of a polymatroidal ideal

L\subset \mathbb {Z}^n

Monomial Ideals with Primary Components given by Powers of Monomial Prime Ideals

L⊂Zn be a lattice and