An Early History of Toric Ideals

TL;DR

This paper provides a personal historical overview of the development of toric ideals and their Gr"obner bases during the early 1990s, highlighting key contributions and applications.

Contribution

It offers a subjective recounting of the first decade of toric ideals, emphasizing Bernd Sturmfels' influential role in their development and promotion.

Findings

Early results established foundational properties of toric ideals.

Significant progress in understanding Gr"obner bases for toric ideals.

Applications in algebraic geometry and combinatorics expanded during this period.

Abstract

Toric ideals are everywhere. They have been in the commutative algebra lexicon since about 1990 when Bernd Sturmfels used the term. The early days of toric ideals and their Gr\"obner bases were full of new results and promising developments in their applications. Bernd has been consistently their biggest promoter through his own work and that of his collaborators and students. This article is a personal and subjective recalling of the first decade of toric ideals when Bernd played a central role.