# Canonical Series Tutorial -- Computing series solutions to regular holonomic systems

If D/I is a regular holonomic D-module, 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 D-module. 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 D-ideal 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.

 i1 : needsPackage "Dmodules" o1 = Dmodules o1 : Package i2 : R1 = QQ[z] o2 = R1 o2 : PolynomialRing i3 : W1 = makeWA R1 o3 = W1 o3 : PolynomialRing, 1 differential variables i4 : a=1/2 1 o4 = - 2 o4 : QQ i5 : b=3 o5 = 3 i6 : c=5/3 5 o6 = - 3 o6 : QQ i7 : J = ideal(z*(1-z)*dz^2+(c-(a+b+1)*z)*dz-a*b) -- the Gauss hypergeometric equation, exponents 0, 1-c 2 2 2 9 5 3 o7 = ideal(- z dz + z*dz - -z*dz + -dz - -) 2 3 2 o7 : Ideal of W1 i8 : cssExpts(J,{1}) 2 o8 = {{0}, {- -}} 3 o8 : List i9 : inw(J,{-1,1}) 2 o9 = ideal(6z*dz + 10dz) o9 : Ideal of W1 i10 : distraction(oo,QQ[s]) 2 o10 = ideal(6s + 4s) o10 : Ideal of QQ[s] i11 : factor oo_0 o11 = (s)(3s + 2)(2) o11 : Expression of class Product i12 : c=1 -- Now we have a single exponent of multiplicity 2 o12 = 1 i13 : J = ideal(z*(1-z)*dz^2+(c-(a+b+1)*z)*dz-a*b) 2 2 2 9 3 o13 = ideal(- z dz + z*dz - -z*dz + dz - -) 2 2 o13 : Ideal of W1 i14 : cssExpts(J,{1}) o14 = {{0}} o14 : List i15 : cssExptsMult(J,{1}) o15 = {{2, {0}}} o15 : List

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]

 i16 : R2 = QQ[x_1..x_3] o16 = R2 o16 : PolynomialRing i17 : W2 = makeWA R2 o17 = W2 o17 : PolynomialRing, 3 differential variables i18 : gens W2 o18 = {x , x , x , dx , dx , dx } 1 2 3 1 2 3 o18 : List i19 : thetaRing = QQ[t_1,t_2,t_3] -- sets variable names t_i = x_i\dx_i o19 = thetaRing o19 : PolynomialRing i20 : f1= x_1*dx_1 -- this element already belongs to thetaRing o20 = x dx 1 1 o20 : W2 i21 : genToDistractionGens(f1,thetaRing) -- checks out o21 = t 1 o21 : thetaRing i22 : f2 = x_1^3*dx_1^3 -- this is a descending factorial in the theta variables 3 3 o22 = x dx 1 1 o22 : W2 i23 : genToDistractionGens(f2,thetaRing) 3 2 o23 = t - 3t + 2t 1 1 1 o23 : thetaRing i24 : factor oo -- now it looks like a descending factorial o24 = (t )(t - 2)(t - 1) 1 1 1 o24 : Expression of class Product i25 : f = x_1^2*x_2^2*x_3*dx_1*dx_2^2*dx_3^2 2 2 2 2 o25 = x x x dx dx dx 1 2 3 1 2 3 o25 : W2 i26 : genToDistractionGens(f,thetaRing) 2 2 2 2 o26 = t t t - t t t - t t t + t t t 1 2 3 1 2 3 1 2 3 1 2 3 o26 : thetaRing

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].

 i27 : A = matrix{{1,1,1},{0,1,2}} o27 = | 1 1 1 | | 0 1 2 | 2 3 o27 : Matrix ZZ <--- ZZ i28 : I = gkz(A,{10,8}) 2 o28 = ideal (x D + x D + x D - 10, x D + 2x D - 8, - D + D D ) 1 1 2 2 3 3 2 2 3 3 2 1 3 o28 : Ideal of QQ[x ..x , D ..D ] 1 3 1 3 i29 : holonomicRank(I) o29 = 2 i30 : cssExpts(I,{1,0,0}) o30 = {{2, 8, 0}, {0, 12, -2}} o30 : List

In this case, the series corresponding to the exponent (2,8,0) is logarithm-free (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.