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On the dimension of the strongly robust complex for configurations in general position

Dimitra Kosta, Apostolos Thoma, Marius Vladoiu

TL;DR

This paper analyzes the strongly robust property of toric ideals through the lens of the strongly robust complex $\Delta_T$ associated to a bouquet ideal $I_T$. It proves that for configurations in general position, $\dim(\Delta_T) < \operatorname{rank}(T)$, providing a positive resolution of Sullivant's question in this setting and showing that such configurations cannot themselves be strongly robust. It further demonstrates how to construct families of strongly robust toric ideals with bouquet ideals arising from general-position configurations via generalized Lawrence matrices, including explicit cyclic configurations that realize the maximal possible $\dim(\Delta_T)$ for a given rank. These results connect combinatorial bouquet structures with explicit algebraic constructions, enriching both the theory of toric ideals and methods for generating robust examples.

Abstract

Strongly robust toric ideals are the toric ideals for which the set of indispensable binomials is the Graver basis. The strongly robust simplicial complex $Δ_T$ of a simple toric ideal $I_T$ determines the strongly robust property for all toric ideals that have $I_T$ as their bouquet ideal. We prove that $\text{dim} Δ_T<\text{rank}(T)$ for configurations in general position, partially answering a question posed by Sullivant.

On the dimension of the strongly robust complex for configurations in general position

TL;DR

This paper analyzes the strongly robust property of toric ideals through the lens of the strongly robust complex associated to a bouquet ideal . It proves that for configurations in general position, , providing a positive resolution of Sullivant's question in this setting and showing that such configurations cannot themselves be strongly robust. It further demonstrates how to construct families of strongly robust toric ideals with bouquet ideals arising from general-position configurations via generalized Lawrence matrices, including explicit cyclic configurations that realize the maximal possible for a given rank. These results connect combinatorial bouquet structures with explicit algebraic constructions, enriching both the theory of toric ideals and methods for generating robust examples.

Abstract

Strongly robust toric ideals are the toric ideals for which the set of indispensable binomials is the Graver basis. The strongly robust simplicial complex of a simple toric ideal determines the strongly robust property for all toric ideals that have as their bouquet ideal. We prove that for configurations in general position, partially answering a question posed by Sullivant.

Paper Structure

This paper contains 4 sections, 4 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Configurations in general position and the dimension of the strongly robust complex
  4. Toric ideals of Cyclic configurations as bouquet ideals

Key Result

Proposition 2.2

Let $T$ be a simple configuration and ${\bf u},{\bf v},{\bf w}\in\operatorname{Ker}_{\mathbb{Z}}(T)$. If $D_{\omega}({\bf u})=D_{\omega}({\bf v})+_{sc}D_{\omega}({\bf w})$ in $\operatorname{Ker}_{\mathbb{Z}}\left(\Lambda (T)_{\omega}\right)$, then $[{\bf u}]^{\omega}=[{\bf v}]^{\omega}+_{c}[{\bf w}]

Theorems & Definitions (13)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Example 2.5
  • Definition 3.1
  • Theorem 3.2
  • proof
  • ...and 3 more