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# polynomialSolutions -- polynomial solutions of a holonomic system

## Synopsis

• Usage:
polynomialSolutions I
polynomialSolutions M
polynomialSolutions(I,w)
polynomialSolutions(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. 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