Gröbner bases of toric ideals and their application

Abstract

The theory of Gröbner bases has a lot of application in many research areas, and is implemented in various mathematical software; see, e.g., [2, 3]. Among their application, this tutorial will focus on basic and recent developments in the theory of Gröbner bases of toric ideals. Toric ideals have been studied for a long time. For example, in the book [9], Herzog's paper [6] was introduced as an early reference. In 1990's, several breakthroughs on toric ideals were done: • Conti--Traverso algorithm for integer programming using Gröbner bases of toric ideals (see [1]); • Correspondence between regular triangulations [5] of integral convex polytopes and Gröbner bases of toric ideals (see [8]); • Diaconis--Sturmfels algorithm for Markov chain Monte Carlo method in the examination of a statistical model using a set of generators of toric ideals (see [4]). In this tutorial, starting with introduction to Gröbner bases and toric ideals, we study some topics related with breakthroughs above. A lot of mathematical software contributed to developments of this research area. (One can find a partial list of such software in Chapters 3 and 7 of [7].)