# gkz -- GKZ A-hypergeometric ideal

## Synopsis

• Usage:
gkz(A,b)
• Inputs:
• Outputs:
• an ideal, which represents the Gel'fand-Kapranov-Zelevinsky hypergeometric system associated to the matrix A and the parameter vector b

## Description

The GKZ hypergeometric system of PDE's associated to a (d x n) integer matrix A consists of the toric ideal I_A in the polynomial subring C[d_1,...,d_n] and Euler relations given by the entries of the vector (A theta - b), where theta is the vector (theta_1,...,theta_n)^t, and theta_i = x_i d_i. See the book 'Groebner deformations of hypergeometric differential equations' by Saito-Sturmfels-Takayama (1999) for more details.
 ```i1 : A = matrix{{1,1,1},{0,1,2}} o1 = | 1 1 1 | | 0 1 2 | 2 3 o1 : Matrix ZZ <--- ZZ``` ```i2 : b = {3,4} o2 = {3, 4} o2 : List``` ```i3 : I = gkz (A,b) 2 o3 = ideal (D - D D , x D + x D + x D - 3, x D + 2x D - 4) 2 1 3 1 1 2 2 3 3 2 2 3 3 o3 : Ideal of QQ[x , x , x , D , D , D ] 1 2 3 1 2 3```

## Caveat

gkz always returns a different ring and will use variables x_1,...,x_n, D_1,...D_n.