5. Convex Polytopes and Gröbner Bases

Gröbner bases of toric ideals have applications in many research areas. Among them, one of the most important topics is the correspondence to triangulations of convex polytopes. It is very interesting that, not only do Gröbner bases give triangulations, but also “good” Gröbner bases give “good” triangulations (unimodular triangulations). On the other hand, in order to use polytopes to study Gröbner bases of ideals of polynomial rings, we need the theory of Gröbner fans and state polytopes. The purpose of this chapter is to explain these topics in detail. First, we will explain convex polytopes, weight vectors, and monomial orders, all of which play a basic role in the rest of this chapter. Second, we will study the Gröbner fans of principal ideals, homogeneous ideals, and toric ideals; this will be useful when we analyze changes of Gröbner bases. Third, we will discuss the correspondence between the initial ideals of toric ideals and triangulations of convex polytopes, and the related ring-theoretic properties. Finally, we will consider the examples of configuration matrices that arise from finite graphs or contingency tables, and we will use them to verify the theory stated above. If you would like to pursue this topic beyond what is included in this chapter, we suggest the books [2, 7].