Abanicos de Groebner e ideales tóricos

Abstract

The Groebner fan of a polynomial ideal was originally introduced by T. Mora and L. Robbiano in 1988. In this Bachelor’s Degree Final Project we study the Groebner fan of a polynomial ideal, including the proof by Fukuda, Jensen and Thomas that it is a polyhedral complex consisting of polyhedral cones. Next, we will see a constructive proof of the existence of the state polytope of a homogeneous ideal, whose normal fan is the Grobner fan of such ideal. On the other hand, we will introduce toric ideals associated with integer matrices and regular triangulations of the latter. We will continue by studying the secondary fan of a toric ideal, whose maximal dimension cones correspond to regular triangulations of the associated matrix. Moreover, we will see that the Grobner fan of a toric ideal is a polyhedral subdivision of its secondary fan. Finally, we will focus on the existence of the secondary polytope of a homogeneous toric ideal, whose normal fan is the secondary fan. Regular triangulations of a matrix, the secondary fan of a toric ideal and its secondary polytope were originally introduced by Gelfand, Kapranov and Zelevinsky in 1989.