# gkz -- The A-hypergeometric systems of Gelfand, Kapranov and Zelevinsky (GKZ)

## Synopsis

• Usage:
gkz(A,b)
gkz(A,b,D)
• Inputs:
• A, ,
• b, a list,
• D, ,
• Outputs:
• an ideal, the GKZ hypergeometric system associated to the matrix A and the parameter vector b in the Weyl algebra D.

## Description

The GKZ hypergeometric system of PDE's associated to a $d$ $\times$ $n$ integer matrix A is an ideal in the Weyl algebra $D_n$ over &#x2102; with generators $x_1,\dots,x_n$ and $\partial_1,\dots,\partial_n$. It consists of the toric ideal $I_A$ in the polynomial subring &#x2102;$[\partial_1,...,\partial_n]$ and Euler relations given by the entries of the vector (A $\theta$ - b), where $\theta$ is the vector $(\theta_1,...,\theta_n)^t$, and $\theta_i = x_i \partial_i$. A field of characteristic zero may be used instead of &#x2102;. For more details, see [SST, Chapters 3 and 4].

 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 (x D + x D + x D - 3, x D + 2x D - 4, - D + D D ) 1 1 2 2 3 3 2 2 3 3 2 1 3 o3 : Ideal of QQ[x ..x , D ..D ] 1 3 1 3 i4 : describe ring I o4 = QQ[x ..x , D ..D , Degrees => {6:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1, WeylAlgebra => {x => D , x => D , x => D }] 1 3 1 3 {GRevLex => {6:1} } 1 1 2 2 3 3 {Position => Up }

The ambient Weyl algebra can be determined as an input.

 i5 : D = makeWA(QQ[x_1..x_3]) o5 = D o5 : PolynomialRing, 3 differential variables i6 : gkz(A,b,D) 2 o6 = ideal (x dx + x dx + x dx - 3, x dx + 2x dx - 4, - dx + dx dx ) 1 1 2 2 3 3 2 2 3 3 2 1 3 o6 : Ideal of D

One may separately produce the toric ideal and the Euler operators.

 i7 : toricIdealPartials(A,D) 2 o7 = ideal(- dx + dx dx ) 2 1 3 o7 : Ideal of QQ[dx ..dx ] 1 3 i8 : eulerOperators(A,b,D) o8 = {x dx + x dx + x dx - 3, x dx + 2x dx - 4} 1 1 2 2 3 3 2 2 3 3 o8 : List

## Caveat

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