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This solution also works:

DEN = prod(d.denominator() for d in list(MolienInt.op))

Wow! Once again I thank you very much!

I will hopefully try this code tomorrow.

Hi,

I am trying to run the following code (I am including the full code, this is about representation theory but my question is much simpler):

B2 = WeylCharacterRing(['B', 2])
b2 = WeightRing(B2)
Weyl = B2(2, 2)

l = B2.rank()
z = [var('z0')]
for k in range(1, l):
    z.append(var('z' + str(k)))
u = var('u')

def toPoly(v):
    return prod([z[k]^v[k] for k in range(0, l)])

Molien = 1
W = Weyl.weight_multiplicities()
for w in W:
    T = toPoly(-w)
    Molien *= 1/(1-u*T)^W[w]

Molien = factor(Molien)
WeylDenom = prod(1-toPoly(-alpha) for alpha in b2.positive_roots())

MolienInt = factor(Molien*WeylDenom/prod(z[k] for k in range(0, l)))
DEN = MolienInt.denominator()
(...)

Everything works fine except for the very last line that seems to run forever. Yet it is pretty straightforward to get the denominator of MolienInt.

I made some tests using PolynomialRing but it also seems not to end and has also the drawback that the denominator is expanded while I need it to remain factored.

Can you help me?

Thank you very much for your answer and also for providing me something almost automated.

You need to remove the mult == 1 in the first computation to get the correct result.

The initial problem is to compute the Hilbert series of the ring of invariants of a certain representation of the orthogonal group. Here the test is for the standard representation of SO(4).

Thank you very much for your answer!

The Cauchy residue theorem was actually my first guess but since I will work with more complicated things that might take me out of the workd of rational functions, I prefered to leave Sage choose the best option.

There is indeed something fishy with the integrate function. You can actually set t equal to any value in the range (0, 1) before evaluation the integral and you'll get the corresponding value $-4\pi^2/t^2$...

Once again thank you very much!

Hi,

Let f be the following function:

f(t, z0, z1) = (z0z1 - 1)(z0 - z1)/((tz0 - 1)(tz1 - 1)(t - z0)(t - z1)z0)

f is a nice holomorphic function and I would like to compute the contour integral

\int_{|z1|=1} \int_{|z0|=1} f(t, z0, z1) dz0/z0 dz1/z1

Assuming 0 < t < 1, the result should be -4 pi^2/(t^2-1).

In Sage:

var("z0, z1, theta0, theta1")

assume(0 < t < 1)

f = (z0z1 - 1)(z0 - z1)/((tz0 - 1)(tz1 - 1)(t - z0)(t - z1)z0)

g = f.subs({z0: exp(Itheta0), z1: exp(Itheta1)})

Now if I integrate with respect to theta0 first: g.integrate(theta0, 0, 2pi) Sage answers that the integral is zero. If I integrate with respect to theta1 first: factor(g.integrate(theta1, 0, 2pi).integrate(theta0, 0, 2pi)) Sage answers that the integral is -4pi^2/t^2 which is also clearly wrong...

Maple finds the right answer.

What can I do to make Sage compute it right? (This is a test I need to compute much more complicated integrals after this so I have to make sure Sage gives me the right answer).