The GKZ hypergeometric system of PDE’s associated to a d × n integer matrix A is an ideal in the Weyl algebra Dn over ℂ with generators x1,…,xn and ∂1,…,∂n. It consists of the toric ideal IA in the polynomial subring ℂ[∂1,...,∂n] and Euler relations given by the entries of the vector (A θ - b), where θ is the vector (θ1,...,θn)t, and θi = xi ∂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 , x , D , D , D ] 1 2 3 1 2 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 , dx ] 1 2 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 |
gkz(A,b) always returns a different ring and will use variables x1,...,xn, D1,...Dn.