# holonomicRank -- rank of a D-module

## Synopsis

• Usage:
holonomicRank M
holonomicRank I
• Inputs:
• M, , over the Weyl algebra D
• I, an ideal, which represents the module M = D/I
• Outputs:

## 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 &#x2102;. and let $R$ denote the ring of differential operators &#x2102;$(x_1,\dots,x_n)<\partial_1,\dots,\partial_n>$ 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 &#x2102;$[x_1,\dots,x_n]$. 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 variables 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