Details:
Title  Convex polytopes and Gr\"obner bases.  Author(s)  Hidefumi Ohsugi  Type  Book, Chapter in Book, Conference Proceeding  Abstract  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 ringtheoretic 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].  ISBN  9784431545736/hbk; 97844 
URL 
http://link.springer.com/chapter/10.1007%2F9784431545743_5 
Language  English  Pages  223278  Publisher  Tokyo: Springer  Year  2013  Edition  0  Translation 
No  Refereed 
No 
