Mathematics, Statistics, and Computer Science
symbolic power, monomial ideals, edge ideal, resurgence, graphs, odd cycle
Given a nontrivial homogeneous ideal I ⊆ k[x1, x2, . . . ,xd], a problem of great recent interest has been the comparison of the rth ordinary power of I and the mth symbolic power I(m). This comparison has been undertaken directly via an exploration of which exponents m and r guarantee the subset containment I(m) ⊆ Ir and asymptotically via a computation of the resurgence ρ(I), a number for which any m/r > ρ(I) guarantees I(m) ⊆ Ir. Recently, a third quantity, the symbolic defect, was introduced; as It ⊆ I(t), the symbolic defect is the minimal number of generators required to add to It in order to get I(t). We consider these various means of comparison when I is the edge ideal of certain graphs by describing an ideal J for which I(t) = It + J. When I is the edge ideal of an odd cycle, our description of the structure of I(t) yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.
Source Publication Title
Journal of Algebra and Its Applications
World Scientific Publishing Company
Janssen, M., Kamp, T., & Vander Woude, J. (2018). Comparing Powers of Edge Ideals. Journal of Algebra and Its Applications, 18 (10), 1950184. https://doi.org/10.1142/S0219498819501846