Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Depth of powers of integrally closed edge ideals of edge-weighted paths

Jiaxin Li, Thanh Vu, Guangjun Zhu

TL;DR

This paper determines the depth of powers of edge ideals of integrally closed edge-weighted paths, showing that the usual limit-depth behavior for unweighted graphs can fail in the weighted setting. Using colon-ideal techniques and inductive arguments, it treats two canonical weight patterns: a single non-trivial weight and two non-trivial weights at positions $a$ and $a+2$, deriving precise formulas for $\operatorname{depth}(S/I(P({\mathbf w}))^t)$ and establishing sharp bounds. The main results yield exact depth formulas and a limit depth of $2$ in the two-non-trivial-weights case, with explicit exceptional cases depending on $a$, $t$, and $n$. The paper also notes that non-integrally closed weight sequences can yield smaller limit depths, highlighting the role of integrally closedness in the results' sharpness and applicability.

Abstract

We compute the depth of powers of edge ideals of integrally closed edge-weighted paths.

Depth of powers of integrally closed edge ideals of edge-weighted paths

TL;DR

This paper determines the depth of powers of edge ideals of integrally closed edge-weighted paths, showing that the usual limit-depth behavior for unweighted graphs can fail in the weighted setting. Using colon-ideal techniques and inductive arguments, it treats two canonical weight patterns: a single non-trivial weight and two non-trivial weights at positions and , deriving precise formulas for and establishing sharp bounds. The main results yield exact depth formulas and a limit depth of in the two-non-trivial-weights case, with explicit exceptional cases depending on , , and . The paper also notes that non-integrally closed weight sequences can yield smaller limit depths, highlighting the role of integrally closedness in the results' sharpness and applicability.

Abstract

We compute the depth of powers of edge ideals of integrally closed edge-weighted paths.
Paper Structure (4 sections, 19 theorems, 61 equations)

This paper contains 4 sections, 19 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The case of a weight sequence with one non-trivial weight
  4. The case of weight sequences with two non-trivial weights

Key Result

Theorem 1.1

Let ${\mathbf w} = (1,\ldots,1,\omega,1,\ldots,1)$, where $\omega > 1$ appears in the $a$-th position, for some $1\le a\le \lfloor\frac{n}{2}\rfloor$. Then, for any $t\ge 2$,

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • proof
  • ...and 23 more