D-modules are modules over rings of differential operators over algebraic varieties. This package is mostly concerned with computations in the *Weyl algebra*, the ring of differential operators over affine space (over a field of characteristic zero). Most algorithms in this package can be found in the book *Gr\"obner deformations of Hypergeometric Differential Equations* by Saito, Sturmfels and Takayama, hereafter referred to as [SST]. This is also the best place to learn about computational D-module theory. The book *Computational Algebraic Geometry with Macaulay2* has a chapter on D-modules and local cohomology. A good introduction to D-module theory is *A primer of algebraic D-modules* by Coutinho.

The Weyl algebra $D_n$ is the free associative algebra in $2n$ variables $x_1,\dots,x_n$, $\partial_1,\dots,\partial_n$, subject to the following relations: the $x$'s commute with each other; the $\partial$'s commute with each other; $x_i$ commutes with $\partial_j$ if $i\neq j$; and finally, $\partial_i x_i = x_i \partial_i +1$ (the Leibniz rule).

i1 : D1 = QQ[z,dz, WeylAlgebra=>{z=>dz}] o1 = D1 o1 : PolynomialRing, 1 differential variables |

As a reality check, let us confirm that this is not a commutative ring. Here is the Leibniz rule.

i2 : dz*z o2 = z*dz + 1 o2 : D1 |

In order to type less, we can use the shortcuts makeWeylAlgebra or makeWA.

i3 : R = QQ[x_1..x_4] o3 = R o3 : PolynomialRing |

i4 : D4 = makeWA R o4 = D4 o4 : PolynomialRing, 4 differential variables |

i5 : describe D4 o5 = QQ[x ..x , dx ..dx , Degrees => {8:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1, WeylAlgebra => {x => dx , x => dx , x => dx , x => dx }] 1 4 1 4 {GRevLex => {8:1} } 1 1 2 2 3 3 4 4 {Position => Up } |

Elements and ideals are handled in the usual Macaulay2 way. Let us look at the Gauss hypergeometric equation for parameters $a=1, b=2, c=3$.

i6 : use D1 o6 = D1 o6 : PolynomialRing, 1 differential variables |

i7 : a = 1, b = 2, c = 3 o7 = (1, 2, 3) o7 : Sequence |

i8 : g = z*(1-z)*dz^2 + (c-(a+b+1)*z)*dz -a*b 2 2 2 o8 = - z dz + z*dz - 4z*dz + 3dz - 2 o8 : D1 |

i9 : I = ideal g 2 2 2 o9 = ideal(- z dz + z*dz - 4z*dz + 3dz - 2) o9 : Ideal of D1 |

The holonomicRank function computes the dimension of the solution space of a linear system of differential equations.

i10 : holonomicRank I o10 = 2 |

A-Hypergeometric systems, also known as GKZ systems (see [SST, Chapters 3 and 4]) are implemented.

i11 : use D4 o11 = D4 o11 : PolynomialRing, 4 differential variables |

i12 : A = matrix{{1,1,1,1},{0,1,3,4}} o12 = | 1 1 1 1 | | 0 1 3 4 | 2 4 o12 : Matrix ZZ <--- ZZ |

i13 : b = {1,2} o13 = {1, 2} o13 : List |

i14 : H = gkz(A,b, D4) o14 = ideal (x dx + x dx + x dx + x dx - 1, x dx + 3x dx + 4x dx - 2, 1 1 2 2 3 3 4 4 2 2 3 3 4 4 ----------------------------------------------------------------------- 3 2 2 2 3 2 - dx + dx dx , dx dx - dx dx , - dx dx + dx dx , - dx + dx dx ) 3 2 4 1 3 2 4 2 3 1 4 2 1 3 o14 : Ideal of D4 |

Holonomic D-ideals are analogous to zero-dimensional ideals in polynomial rings (see [SST, Section 1.4]). Let us confirm that our GKZ systems are holonomic.

i15 : isHolonomic H o15 = true |

Once we know our ideal is holonomic, we can compute its holonomic rank. The above is a famous GKZ example because the holonomic rank may change when the parameter vector $b$ is changed.

i16 : holonomicRank H o16 = 5 |

i17 : holonomicRank sub(gkz(A,{1,0}), vars D4) o17 = 4 |

We can compute the characteristic ideal and singular locus of a D-ideal [SST, Section 1.4]. Note that the output of charIdeal belongs to a commutative ring, the associated graded ring of $D_n$ with respect to the order filtration.

i18 : charIdeal H o18 = ideal (dx dx - dx dx , x dx + 3x dx + 4x dx , x dx - 2x dx - 2 3 1 4 2 2 3 3 4 4 1 1 3 3 ----------------------------------------------------------------------- 3 2 2 2 3 2 3x dx , dx - dx dx , dx dx - dx dx , dx - dx dx ) 4 4 3 2 4 1 3 2 4 2 1 3 o18 : Ideal of QQ[x ..x , dx ..dx ] 1 4 1 4 |

i19 : singLocus H 3 3 3 4 2 2 2 2 4 3 3 3 o19 = ideal(4x x x x + 27x x x + 6x x x x + 27x x x + 192x x x x - 1 2 3 4 1 3 4 1 2 3 4 1 2 4 1 2 3 4 ----------------------------------------------------------------------- 4 4 256x x ) 1 4 o19 : Ideal of D4 |

The singular locus of a GKZ system is the zero set of a polynomial called the Principal A-determinant, which is a product of discriminants corresponding to faces of the matrix A (see Chapters 8 and 9 of the book *Discriminants, Resultants and Multidimensional Determinants* by Gelfand, Kapranov and Zelevinsky). Here is how to find the classic cubic discriminant.

i20 : A1 = matrix{{1,1,1,1},{0,1,2,3}}, b1={0,0} o20 = (| 1 1 1 1 |, {0, 0}) | 0 1 2 3 | o20 : Sequence |

i21 : H1 = sub(gkz(A1,b1),vars D4) 2 o21 = ideal (x dx + x dx + x dx + x dx , x dx + 2x dx + 3x dx , - dx + 1 1 2 2 3 3 4 4 2 2 3 3 4 4 3 ----------------------------------------------------------------------- 2 dx dx , - dx + dx dx , - dx dx + dx dx ) 2 4 2 1 3 2 3 1 4 o21 : Ideal of D4 |

i22 : factor (singLocus H1)_0 2 2 3 3 2 2 o22 = (x )(x )(x x - 4x x - 4x x + 18x x x x - 27x x ) 4 1 2 3 1 3 2 4 1 2 3 4 1 4 o22 : Expression of class Product |

- Dmodules -- algorithms for D-modules