# 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, ,
• thetaRing, , a stand in for the subring $k[\theta]$ generated by theta = x*dx
• Outputs:
• an ideal or , 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 .