Vanishing ideals over complete multipartite graphs

TL;DR

This paper explicitly describes the generators of the vanishing ideal associated with a complete multipartite graph over a finite field, and determines its Castelnuovo–Mumford regularity based on these generators.

Contribution

It provides an explicit family of binomial generators for the ideal and computes its regularity, advancing understanding of algebraic properties of graph-associated toric ideals.

Findings

Explicit binomial generators for the ideal are given.

The regularity of the ideal is computed explicitly.

Generators include those from the torus, 4-cycle quadratics, and degree q-1 binomials.

Abstract

We study the vanishing ideal of the parametrized algebraic toric associated to the complete multipartite graph $\G=\mathcal{K}_{\alpha_1,...,\alpha_r}$ over a finite field of order $q$. We give an explicit family of binomial generators for this lattice ideal, consisting of the generators of the ideal of the torus, (referred to as type I generators), a set of quadratic binomials corresponding to the cycles of length 4 in $\G$ and which generate the \emph{toric algebra of $\G$} (type II generators) and a set of binomials of degree $q-1$ obtained combinatorially from $\G$ (type III generators). Using this explicit family of generators of the ideal, we show that its Castelnuovo--Mumford regularity is equal to $\max\set{\alpha_1(q-2),...,\alpha_r(q-2), \lceil (n-1)(q-2)/2\rceil}$, where $n=\alpha_1+... + \alpha_r$.