toricIdealPartials -- create the toric ideal of an integer matrix

Synopsis

Usage:

toricIdealPartials(A,D)
Inputs:
- A, a matrix,
- D, a polynomial ring,
Outputs:
- an ideal, the toric ideal of the matrix $A$ in the polynomial ring of the partials inside of the Weyl algerba $D$.

Description

A $d \times n$ integer matrix $A$ determines a GKZ hypergeometric system of PDEs in the Weyl algebra $D_n$ over $\mathbb{C}$. The matrix $A$ is associated to the toric ideal $I_A$ in the polynomial subring $\mathbb{C}[\partial_1,...,\partial_n]$ of $D$. A field of characteristic zero may be used instead of $\mathbb{C}$. For more details, see [SST, Chapters 3 and 4].

i1 : A = matrix{{1,2,0},{-1,1,3}}

o1 = | 1  2 0 |
     | -1 1 3 |

              2       3
o1 : Matrix ZZ  <-- ZZ

i2 : D = makeWA(QQ[x_1..x_3])

o2 = D

o2 : PolynomialRing, 3 differential variable(s)

i3 : I = toricIdealPartials(A,D)

             2
o3 = ideal(dx dx  - dx )
             1  3     2

o3 : Ideal of QQ[dx ..dx ]
                   1    3

i4 : describe ring I

o4 = QQ[dx ..dx , Degrees => {3:1}, Heft => {1}]
          1    3

Ways to use toricIdealPartials :

toricIdealPartials(Matrix,PolynomialRing)

For the programmer

The object toricIdealPartials is a method function.