arXiv subjects

Ha Minh Lam

Publications and source records attributed to Ha Minh Lam.

6 recordsLinked to original sources

arXiv2024

A general formula for the index of depth stability of edge ideals

By a classical result of Brodmann, the function $\operatorname{depth} R/I^t$ is asymptotically a constant, i.e. there is a number $s$ such that $\operatorname{depth} R/I^t = \operatorname{depth} R/I^s$ for $t > s$. One calls the smallest number $s$ with this property the index of depth stability of $I$ and denotes it by $\operatorname{dstab}(I)$. This invariant remains mysterious til now. The main result of this paper gives an explicit formula for $\operatorname{dstab}(I)$ when $I$ is an arbitrary ideal generated by squarefree monomials of degree 2. That is the first general case where one can characterize $\operatorname{dstab}(I)$ explicitly. The formula expresses $\operatorname{dstab}(I)$ in terms of the associated graph. The proof involves new techniques which relate different topics such as simplicial complexes, systems of linear inequalities, graph parallelizations, and ear decompositions. It provides an effective method for the study of powers of edge ideals.

math.AC↗

arXiv2023

Lojasiewicz inequalities in a certain class of smooth functions

Let $f$ be a germ of a smooth function at the orirgin in $\RR^n.$ We show that if $f$ is Kouchnirenko's nondegenerate and satisfies the so called Kamimoto--Nose condition then it admits the Łojasiewicz inequalities. We compute the Łojasiewicz exponents for some special cases. In particular, if $f$ is a germ of a smooth convex Kouchnirenko's nondegenerate function and satisfies the Kamimoto--Nose condition, then all its Łojasiewicz exponents can be expressed very simply in terms of its Newton polyhedron.

math.AG↗

arXiv2023

Decreasing behavior of the depth functions of edge ideals

Let $I$ be the edge ideal of a connected non-bipartite graph and $R$ the base polynomial ring. Then $\operatorname{depth} R/I \ge 1$ and $\operatorname{depth} R/I^t = 0$ for $t \gg 1$. We give combinatorial conditions for $\operatorname{depth} R/I^t = 1$ for some $t$ in between and show that the depth function is non-increasing thereafter. Especially, the depth function quickly decreases to 0 after reaching 1. We show that if $\operatorname{depth} R/I = 1$ then $\operatorname{depth} R/I^2 = 0$ and if $\operatorname{depth} R/I^2 = 1$ then $\operatorname{depth} R/I^5 = 0$. Other similar results suggest that if $\operatorname{depth} R/I^t = 1$ then $\operatorname{depth} R/I^{t+3} = 0$. This a surprising phenomenon because the depth of a power can determine a smaller depth of another power. Furthermore, we are able to give a simple combinatorial criterion for $\operatorname{depth} R/I^{(t)} = 1$ for $t \gg 1$ and show that the condition $\operatorname{depth} R/I^{(t)} = 1$ is persistent, where $I^{(t)}$ denotes the $t$-th symbolic powers of $I$.

math.AC↗

arXiv2018

Associated primes of powers of edge ideals and ear decompositions of graphs

In this paper, we give a complete description of the associated primes of every power of the edge ideal in terms of generalized ear decompositions of the graph. This result establishes a surprising relationship between two seemingly unrelated notions of Commutative Algebra and Combinatorics. It covers all previous major results in this topic and has several interesting consequences.

math.AC↗

arXiv2015

Combinatorial characterizations of the saturation and the associated primes of the fourth power of edge ideals

To compute the local cohomology of powers of edge ideals one needs to know their saturations. The saturation of the second and third powers has been described in terms of the graph in [13] and [10]. In this article, we give a combinatorial description of the generators of the saturation of the fourth power. As a consequence, we are able to give a complete classification of the associated primes of the fourth power of edge ideals in terms of the graph.

math.AC↗

arXiv2014

Saturation and associated primes of powers of edge ideals

For the edge ideal I of an arbitrary simple graph G we describe the monomials of the saturation of a power of I in terms of (vertex) weighted graphs associated with the monomials. This description allows us to characterize the embedded associated primes of a power of I as covers of G which contain certain types of subgraphs of G. As an application, we completely classify the associated primes of the second and the third power of I in terms of G.

math.AC↗