holonomicRank -- holonomic rank of a D-module

Synopsis

Usage:

holonomicRank M

holonomicRank I
Inputs:
- M, a module, over the Weyl algebra D
- I, an ideal, which represents the module M = D/I
Outputs:
- an integer, the rank of M

Description

The holonomic rank of a D-module M = D^r/N provides analytic information about the system of PDE's given by N. By the Cauchy-Kovalevskii-Kashiwara Theorem, the dimension of the space of germs of holomorphic solutions to N in a neighborhood of a nonsingular point is equal to the holonomic rank of M.

The holonomic rank of a D-module is defined algebraically as follows. Let $D$ be the Weyl algebra with generators $x_1,\dots,x_n$ and $\partial_1,\dots,\partial_n$ over ℂ. and let $R$ denote the ring of differential operators ℂ$(x_1,\dots,x_n)<\partial_1,\dots,\partial_n>$ with rational function coefficients. Then the holonomic rank of $M = D^r/N$ is equal to the dimension of $R^r/RN$ as a vector space over ℂ$[x_1,\dots,x_n]$. More details can be found in [SST, Section 1.4].

i1 : makeWA(QQ[x,y])

o1 = QQ[x..y, dx, dy]

o1 : PolynomialRing, 2 differential variable(s)

i2 : I = ideal (x*dx+2*y*dy-3, dx^2-dy)

                                2
o2 = ideal (x*dx + 2y*dy - 3, dx  - dy)

o2 : Ideal of QQ[x..y, dx, dy]

i3 : holonomicRank I

o3 = 2

Ways to use holonomicRank :

holonomicRank(Ideal)
holonomicRank(Module)

For the programmer

The object holonomicRank is a method function.

holonomicRank -- holonomic rank of a D-module

Synopsis

Description

See also

Ways to use holonomicRank :

For the programmer