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Dmodules :: holonomicRank

holonomicRank -- rank of a D-module

Synopsis

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

See also

Ways to use holonomicRank :

For the programmer

The object holonomicRank is a method function.