next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
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 x1,…,xn and 1,…,∂n over ℂ. and let R denote the ring of differential operators ℂ(x1,…,xn)<∂1,…,∂n> with rational function coefficients. Then the holonomic rank of M = Dr/N is equal to the dimension of Rr/RN as a vector space over ℂ[x1,…,xn]. 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 :