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Toric splittings

Anargyros Katsabekis, Apostolos Thoma

Abstract

The toric ideal $I_A$ is splittable if it has a toric splitting; namely, if there exist toric ideals $I_{A_1}, I_{A_2}$ such that $I_A=I_{A_1}+I_{A_2}$ and $I_{A_i}\not =I_{A}$ for all $1 \leq i \leq 2$. We provide a necessary and sufficient condition for a toric ideal to be splittable in terms of $A$, and we apply it to prove or disprove that certain classes of toric ideals are splittable.

Toric splittings

Abstract

The toric ideal is splittable if it has a toric splitting; namely, if there exist toric ideals such that and for all . We provide a necessary and sufficient condition for a toric ideal to be splittable in terms of , and we apply it to prove or disprove that certain classes of toric ideals are splittable.

Paper Structure

This paper contains 3 sections, 10 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Splitting Criterion for $I_A$
  3. Applications

Key Result

Theorem 2.1

The toric ideal $I_A$ is splittable if and only if there exists a minimal system of binomial generators $\{B({\bf u}) \mid {\bf u}\in C\subset {\rm ker}_{\mathbb{Z}}(A)\}$ of the toric ideal $I_A$, and sets $C_{1}$ and $C_2$ such that $C=C_1 \cup C_2$, ${\rm span}_{\mathbb{Q}}(C_1)\subsetneqq {\rm k

Theorems & Definitions (14)

  • Theorem 2.1
  • Example 2.2
  • Example 2.3
  • Remark 2.4
  • Theorem 2.5
  • Theorem 3.1
  • Example 3.2
  • Corollary 3.3
  • Theorem 3.4
  • Theorem 3.5
  • ...and 4 more