distraction -- the image in the thetaRing of a torus-fixed element or ideal in a Weyl algebra

Synopsis

Usage:

distraction(I,thetaRing)

distraction(f,thetaRing)
Inputs:
- I, an ideal,
- f, a ring element,
- thetaRing, a polynomial ring, a stand in for the subring $k[\theta]$ generated by theta = x*dx
Outputs:
- an ideal or a ring element, that results from intersecting with the thetaRing of D, as in [SST, Lemma 2.3.1] to I.

Description

Given a monomial $x^u \partial^v$, this function rewrites it as a product $x^a p(\theta) \partial^b$, where $\theta_i = x_i \partial_i$ for $i = 1,\dots, n$. This is a step in a procedure for checking that D-ideal is torus-fixed, and is used in the isTorusFixed routine.

Given a torus fixed $D$-ideal, this function computes the distraction as in [SST, Corollary 2.3.5]. This is necessary to compute indicial ideals [SST, Theorem 2.3.9, Corollary 2.3.5]; the roots of the indicial ideals are the exponents of the starting terms of canonical series solutions of regular holonomic D-ideals.

i1 : R1 = QQ[z]

o1 = R1

o1 : PolynomialRing

i2 : W1 = makeWA R1

o2 = W1

o2 : PolynomialRing, 1 differential variable(s)

i3 : a=1/2

     1
o3 = -
     2

o3 : QQ

i4 : b=3

o4 = 3

i5 : c=5/3

     5
o5 = -
     3

o5 : QQ

i6 : 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
o6 = ideal(- z dz  + z*dz  - -z*dz + -dz - -)
                             2       3     2

o6 : Ideal of W1

i7 : cssExpts(J,{1})

              2
o7 = {{0}, {- -}}
              3

o7 : List

i8 : K = inw(J,{-1,1})

                2
o8 = ideal(6z*dz  + 10dz)

o8 : Ideal of W1

i9 : distraction(K,QQ[s])

             2
o9 = ideal(6s  + 4s)

o9 : Ideal of QQ[s]

Ways to use distraction :

distraction(Ideal,Ring)
distraction(RingElement,Ring)

For the programmer

The object distraction is a method function.