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Dmodules :: gkz

gkz -- GKZ A-hypergeometric ideal

Synopsis

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.

See also

Ways to use gkz :