holonomicRank -- holonomic rank of a D-module

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 $\CC$. and let $R$ denote the ring of differential operators $\CC(x_1,\dots,x_n)\langle\partial_1,\dots,\partial_n\rangle$ 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 $\CC[x_1,\dots,x_n]$.

The holonomic rank is also determined by the number of standard monomials $\{\partial^\alpha\}$ with respect to a Gröbner basis of $I$ for any term order on $R$. For convenience, these standard monomials, which form a basis for differential operators modulo the system, are cached.

See [SST, Algorithm 1.4.17] for more details.