next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
Dmodules :: gkz

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

Synopsis

Description

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 1,...,θn)t, and θi = xii. 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

Caveat

gkz(A,b) always returns a different ring and will use variables x1,...,xn, D1,...Dn.

See also

Ways to use gkz :