polynomialSolutions -- polynomial solutions of a holonomic system

Synopsis

Usage:

polynomialSolutions I

polynomialSolutions M

polynomialSolutions(I,w)

polynomialSolutions(M,w)
Inputs:
- M, a module, over the Weyl algebra $D$
- I, an ideal, holonomic ideal in the Weyl algebra $D$
- w, a list, a weight vector
Optional inputs:
- Alg => ..., default value GD, algorithm for finding polynomial solutions
Outputs:
- a list, a basis of the polynomial solutions of $I$ (or of $D$-homomorphisms between $M$ and the polynomial ring) using $w$ for Groebner deformations. If no $w$ is given, then it is taken to be the all ones vector.

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 Gröbner deformations and works for ideals $I$ of PDE's - see the paper Polynomial and rational solutions of a holonomic system by Oaku, Takayama and Tsai (2000). The second algorithm is based on homological algebra - see the paper Computing homomorphisms between holonomic D-modules by Tsai and Walther (2000).

i1 : makeWA(QQ[x])

o1 = QQ[x, dx]

o1 : PolynomialRing, 1 differential variable(s)

i2 : I = ideal(dx^2, (x-1)*dx-1)

              2
o2 = ideal (dx , x*dx - dx - 1)

o2 : Ideal of QQ[x, dx]

i3 : polynomialSolutions I

o3 = {x - 1}

o3 : List

Ways to use polynomialSolutions :

polynomialSolutions(Ideal)
polynomialSolutions(Ideal,List)
polynomialSolutions(Module)
polynomialSolutions(Module,List)

For the programmer

The object polynomialSolutions is a method function with options.

polynomialSolutions -- polynomial solutions of a holonomic system

Synopsis

Description

See also

Ways to use polynomialSolutions :

For the programmer