If D/I is a regular holonomic Dmodule, the solutions of the system of differential equations I can be written as Nilsson series (Puiseux series with logarithms). The constructive version of this result is the canonical series method [SST, Sections 2.5, 2.6]. In this tutorial, we illustrate an implementation of this method.
If the input ideal I is not regular, this method is not guaranteed to produce convergent series, or even holonomicRank(I) formal power series solutions of I. There currently exists no computational method to verify whether D/I is a regular holonomic Dmodule. In the case of GKZ systems, regularity has been characterized in terms of the input matrix.
Contains the following functions:
Currently, this contains the computation of exponents with respect to a weight vector. Completing the canonical series computation is in the future. To compute the exponents for a Dideal I with respect to w, do as follows. Compute the initial ideal of I with respect to w. Introduce the subring of D consisting of polynomials in $\theta_1 = x_1 \partial_1, ... , \theta_n= x_n \partial_n$. This is a commutative polynomial subring of D, referred to here as thetaRing. The indicial ideal of I with respect to w is produced by extending the initial ideal to the ring of differential operators with rational function coefficients, and contract to thetaRing. The exponents of I with respect to w are the roots of the indicial ideal, counted with multiplicities.














The first step is to rewrite elements of the initial ideal in a terms of the thetaRing, in a way that will allow us to easily extend and contract see [SST]











Here is an example that can be continued when more functions are implemented. This is worked out as [page 138, ex 3.5.3, SST].




In this case, the series corresponding to the exponent (2,8,0) is logarithmfree (actually, this is a hypergeometric polynomial), while the series corresponding to (0,12,2) has logarithms. [SST, page 138] has the polynomial, and four terms of the logarithmic series.