# toricIdealPartials -- create the toric ideal of an integer matrix

## Synopsis

• Usage:
toricIdealPartials(A,D)
• Inputs:
• A, ,
• D, ,
• 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 .