Fall 2020 Regularity of the vanishing ideal over a parallel composition of paths
Antonio Macchia, Jorge Neves, Maria Vaz Pinto, Rafael H. Villarreal
J. Commut. Algebra 12(3): 391-407 (Fall 2020). DOI: 10.1216/jca.2020.12.391

Abstract

Let G be a graph obtained by taking r2 paths and identifying all first vertices and identifying all last vertices. We compute the Castelnuovo–Mumford regularity of the quotient SI(X), where S is the polynomial ring on the edges of G and I(X) is the vanishing ideal of the projective toric subset parameterized by G. This invariant is known for several special families of graphs such as trees, cycles, complete graphs and complete bipartite graphs. For bipartite graphs, it is also known that the computation of the regularity can be reduced to the 2-connected case. Thus, we focused on the first case of a bipartite graph where the regularity was unknown. We also prove new inequalities relating the Castelnuovo–Mumford regularity of SI(X) with the combinatorial structure of G, for a general graph.

Citation

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Antonio Macchia. Jorge Neves. Maria Vaz Pinto. Rafael H. Villarreal. "Regularity of the vanishing ideal over a parallel composition of paths." J. Commut. Algebra 12 (3) 391 - 407, Fall 2020. https://doi.org/10.1216/jca.2020.12.391

Information

Received: 23 June 2017; Revised: 23 December 2017; Accepted: 14 January 2018; Published: Fall 2020
First available in Project Euclid: 5 September 2020

zbMATH: 07246826
MathSciNet: MR4146367
Digital Object Identifier: 10.1216/jca.2020.12.391

Subjects:
Primary: 13F20
Secondary: 11T55 , 14G15

Keywords: binomial ideals , Castelnuovo–Mumford regularity , parallel composition of paths

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium

Vol.12 • No. 3 • Fall 2020
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