Persistence and stability properties of powers of ideals

We introduce the concept of strong persistence and show that it implies persistence regarding the associated prime ideals of the powers of an ideal. We also show that strong persistence is equivalent to a condition on power of ideals studied by Ratliff. Furthermore, we give an upper bound for the depth of powers of monomial ideals in terms of their linear relation graph, and apply this to show that the index of depth stability and the index of stability for the associated prime ideals of polymatroidal ideals is bounded by their analytic spread.Comment: 15 pages, 1 figur

Gr\"obner bases of balanced polyominoes

We introduce balanced polyominoes and show that their ideal of inner minors is a prime ideal and has a quadratic Gr\"obner basis with respect to any monomial order, and we show that any row or column convex and any tree-like polyomino is simple and balanced

Regularity of joint-meet ideals of distributive lattices

Let $L$ be a distributive lattice and $R(L)$ the associated Hibi ring. We compute \reg R(L) when $L$ is a planar lattice and give a lower bound for \reg R(L) when $L$ is non-planar, in terms of the combinatorial data of $L.$ As a consequence, we characterize the distributive lattices $L$ for which the associated Hibi ring has a linear resolution

Polarization of Koszul cycles with applications to powers of edge ideals of whisker graphs

In this paper, we introduced the polarization of Koszul cycles and use it to study the depth function of powers of edge ideals of whisker graphs