PolySols -- polynomial solutions of a holonomic system

Synopsis

• Usage:
PolySols I, PolySols M, PolySols(I,w), PolySols(M,w)
• Inputs:
• M, , over the Weyl algebra D
• I, an ideal, holonomic ideal in the Weyl algebra D
• w, a list, a weight vector
• Optional inputs:
• Alg => ..., -- algorithm for finding polynomial solutions
• Outputs:
• a list, a basis of the polynomial solutions of I(or of D-homomorhpisms between M and the polynomial ring) using w for Groebner deformations

Description

The polynomial solutions of a holonomic system form a finite-dimensional vector space. There are two algorithms implemented to get these solutions. The first algorithm is based on Groebner deformations and works for ideals I of PDE's -- see the paper 'Polynomial and rational solutions of a holonomic system' by Oaku-Takayama-Tsai (2000). The second algorithm is based on homological algebra -- see the paper 'Computing homomorphims between holonomic D-modules' by Tsai-Walther (2000).
 ```i1 : W = QQ[x, D, WeylAlgebra=>{x=>D}] o1 = W o1 : PolynomialRing``` ```i2 : I = ideal(D^2, (x-1)*D-1) 2 o2 = ideal (D , x*D - D - 1) o2 : Ideal of W``` ```i3 : PolySols I o3 = {x - 1} o3 : List```

• RatSols -- rational solutions of a holonomic system
• Dintegration -- integration modules of a D-module

Ways to use PolySols :

• PolySols(Ideal)
• PolySols(Ideal,List)
• PolySols(Module)
• PolySols(Module,List)