# 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 => ..., 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

## 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 Gr\"obner 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 homomorphims between holonomic D-modules by Tsai and Walther (2000).

 i1 : makeWA(QQ[x]) o1 = QQ[x, dx] o1 : PolynomialRing, 1 differential variables i2 : I = ideal(dx^2, (x-1)*dx-1) 2 o2 = ideal (dx , x*dx - dx - 1) o2 : Ideal of QQ[x, dx] 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)"

## For the programmer

The object PolySols is .