TIGERS Functions
****************

This chapter describes  interface functions for tigers ox server ox_sm1_tigers.

tigers.tigers
-------------

tigers.tigers(A|proc=A)
     ::   It asks the tigers server of the descriptor number P to enumerate all Grobner bases associated to the toric variaty defined by the matrix A.

return
     List

p
     Number

a
     List

  * It asks the tigers server of the descriptor number P to enumerate all Grobner bases associated to the toric variaty defined by the matrix A.

  * The system tigers is an expert system to enumerate all Gr\"obner bases of affine toric ideals. In other words, it can be used to determine the state polytope of a given affine toric ideal. As to a theoretical background, see the book

     B.Sturmfels, Grobner bases and Convex Polytopes.

     The original tigers is written by Birk Hubert. The algorithm used in explained in the paper

     B.Huber and R.Thomas, Computing Grobner Fans of Toric Ideals.

     [395] A=[[1,1,1,1],[0,1,2,3]]$
     [306] S=tigers.tigers(A)$
     [307] length(S);
     8
     [308] S[0];
     [[[1,0,1,0],[0,2,0,0]],[[1,0,0,1],[0,1,1,0]],[[0,1,0,1],[0,0,2,0]]]
     [309] S[1];
     [[[1,0,0,1],[0,1,1,0]],[[0,2,0,0],[1,0,1,0]],[[0,1,0,1],[0,0,2,0]]]

In this example, all reduced Grobner bases for the toric ideal associated to the matrix A are stored in S. There are eight distinct Grobner bases of A. [[i_1, i_2, ...],[j_1, j_2, ...]] is a set of exponents of two monomials and stands for a binomial. For example, the S[0] consists of

x1 x3 - x2^2,   x1 x4 - x2 x3, x2 x4 - x3^2.

<x1 x3, x1 x4, x2 x4> is the initial ideal of S[0].