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

- 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.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/HolonomicSystems/DOC/canonicalSeries.m2:127:0.