Computing Groebner Fans of Toric Ideals

Birkett Huber and Rekha Thomas
Experimental Mathematics 9 (2000) 321-331.

Abstract

The monomial initial ideals of a graded polynomial ideal are in bijection with the vertices of a convex polytope known as the state polytope of the ideal. The Groebner fan of the ideal is the normal fan of its state polytope. In this paper we present a software system called TiGERS for computing the Groebner fan of a toric ideal by enumerating the edge graph of its state polytope. The key contributions are an inexpensive algorithm for local change of Groebner bases in toric ideals and the identification of a reverse search tree on the vertices of the state polytope. Using these ideas we obtain a combinatorial Groebner walk procedure for toric ideals. TiGERS has been used to compute state polytopes with over 200,000 vertices.

postscript file of the paper

Software: TiGERS (written by Birkett Huber)

TiGERS (uuencoded)

Installing TiGERS:

The distribution is contained in a file named TiGERS_0.9.uu.

To unpack it use the commands:

uudecode TiGERS_0.9.uu
gunzip TiGERS_0.9.taz
tar -xvf TiGERS_0.9.tar

You now have a directory named TiGERS_0.9

The README file now contains further information.