polynomialExt -- Ext groups between a holonomic module and a polynomial ring

Synopsis

• Usage:

  polynomialExt M, polynomialExt I; rationalFunctionExt(i,M), rationalFunctionExt(i,I)

• Inputs:
  • M, a module, over the Weyl algebra D
  • I, an ideal, which represents the module M = D/I
  • i, an integer, nonnegative
• Optional inputs:
  • Strategy => ..., default value Schreyer,
• Outputs:
  • a hash table or a module, the Ext^i group(s) between holonomic M and the polynomial ring

Description

The algorithm used appears in the paper 'Polynomial and rational solutions of holonomic systems' by Oaku-Takayama-Tsai (2000). The method is to combine isomorphisms of Bjork and Kashiwara with the restriction algorithm.

i1 : W = QQ[x, D, WeylAlgebra=>{x=>D}]

o1 = W

o1 : PolynomialRing, 1 differential variable(s)

i2 : M = W^1/ideal(x^2*D^2)

o2 = cokernel | x2D2 |

                            1
o2 : W-module, quotient of W

i3 : polynomialExt(M)

                      2
o3 = HashTable{0 => QQ }
                      2
               1 => QQ

o3 : HashTable

Caveat

See also

• polynomialSolutions -- polynomial solutions of a holonomic system
• rationalFunctionExt -- Ext(holonomic D-module, polynomial ring localized at the singular locus)
• DExt -- Ext groups between holonomic modules
• Dintegration -- integration modules of a D-module