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 ℂ 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 ℂ$[\partial_1,...,\partial_n]$ 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 ℂ. 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 }
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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
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gkz(A,b) always returns a different ring and will use variables x_1,...,x_n, D_1,...D_n.
The object gkz is a method function.