Research ArticleNo Access

Comparing powers of edge ideals

Mike Janssen

Mathematics/Statistics Department, Dordt College, Sioux Center, IA 51250, USA

E-mail Address: mike.janssen@dordt.edu

Corresponding author.

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Thomas Kamp

Mathematics/Statistics Department, Dordt College, Sioux Center, IA 51250, USA

E-mail Address: thmskmp@dordt.edu

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Jason Vander Woude

Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA

E-mail Address: jasonvw@huskers.unl.edu

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https://doi.org/10.1142/S0219498819501846Cited by:5 (Source: Crossref)

Abstract

Given a nontrivial homogeneous ideal $I \subseteq k [x_{1}, x_{2}, \dots, x_{d}]$ $I \subseteq k [x_{1}, x_{2}, \dots, x_{d}]$ , a problem of great recent interest has been the comparison of the $r$ $r$ th ordinary power of $I$ $I$ and the $m$ $m$ th symbolic power $I^{(m)}$ $I^{(m)}$ . This comparison has been undertaken directly via an exploration of which exponents $m$ $m$ and $r$ $r$ guarantee the subset containment $I^{(m)} \subseteq I^{r}$ $I^{(m)} \subseteq I^{r}$ and asymptotically via a computation of the resurgence $ρ (I)$ $ρ (I)$ , a number for which any $m / r > ρ (I)$ $m / r > ρ (I)$ guarantees $I^{(m)} \subseteq I^{r}$ $I^{(m)} \subseteq I^{r}$ . Recently, a third quantity, the symbolic defect, was introduced; as $I^{t} \subseteq I^{(t)}$ $I^{t} \subseteq I^{(t)}$ , the symbolic defect is the minimal number of generators required to add to $I^{t}$ $I^{t}$ in order to get $I^{(t)}$ $I^{(t)}$ . We consider these various means of comparison when $I$ $I$ is the edge ideal of certain graphs by describing an ideal $J$ $J$ for which $I^{(t)} = I^{t} + J$ $I^{(t)} = I^{t} + J$ . When $I$ $I$ is the edge ideal of an odd cycle, our description of the structure of $I^{(t)}$ $I^{(t)}$ yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.

Communicated by T. H. Ha

Keywords:

AMSC: 13F20

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