isTorusFixed -- checks if an ideal in a Weyl algebra is torus-fixed

Synopsis

Usage:

isTorusFixed I
Inputs:
- I, an ideal, in a WeylAlgebra
Outputs:
- a Boolean value, true if I is torus-fixed, false if not

Description

There is a natural action of the n-dimensional algebraic torus on $D$ where $t \in (\mathbb{C}^*)^n$ acts on $\partial_i$ as $t_i\partial_i$ and on $x_i$ as $t_i^{-1}x_i$. The function isTorusFixed verifies whether a D-ideal is invariant under this action.

See [SST], just before Lemma 2.3.1.

i1 : W = makeWA(QQ[x_1,x_2])

o1 = W

o1 : PolynomialRing, 2 differential variable(s)

i2 : b = 2

o2 = 2

i3 : I = ideal(x_1*dx_1*(x_1*dx_1+b), x_1*dx_1*(x_2*dx_2+b),
         x_2*dx_2*(x_1*dx_1+b), x_2*dx_2*(x_2*dx_2+b))

             2  2                                                      2  2
o3 = ideal (x dx  + 3x dx , x x dx dx  + 2x dx , x x dx dx  + 2x dx , x dx  +
             1  1     1  1   1 2  1  2     1  1   1 2  1  2     2  2   2  2  
     ------------------------------------------------------------------------
     3x dx )
       2  2

o3 : Ideal of W

i4 : isTorusFixed I

o4 = true

Ways to use isTorusFixed :

isTorusFixed(Ideal)

For the programmer

The object isTorusFixed is a method function.