fraction field of InfinitePolynomialRing

[chapoton]

I would like to use the fraction field of infinite polynomial ring.

sage: P.<x,y> = InfinitePolynomialRing(QQ, implementation='sparse')
sage: z=P(x[1])
sage: 1/z

So far, one gets

NotImplementedError: Fraction Fields of Infinite Polynomial Rings are not implemented

This seems to be a coercion problem.

[ylchapuy]

Also note that this seems to work (at least a little bit):

sage: X.<x,y> = InfinitePolynomialRing(QQ, implementation='sparse')
sage: F = FractionField(X)
sage: z = F(x[1])
sage: 1/z
1/x_1
sage: (1/z*x[1]).is_one()
True

[chapoton]

It seems now that the example of comment 1 does no longer work.

Here is a proposed patch, that could have worked maybe.

[chapoton]

Here is a patch that works (but there is a problem still with reduction)

[chapoton]

The main problem comes from the gcd :

sage: R.<x>=InfinitePolynomialRing(QQ)
sage: p1=x[0]+x[0]**2
sage: p1.gcd(p1)                      

fails completely.

[chapoton]

well, it almost works now. I have added a custom gcd procedure.

I am not sure that this is the correct way to handle these problems. If somebody has a better idea ?

[tscrim]

I think this is the right way since the gcd needed an implementation. However I have a few comments:

• Could you add some tests showing that you get something in the fraction field?
• Now we don't need the # indirect doctest for any method with a leading and trailing underscore.
• Are you pushing this off to 6.0 because of the large number of changes to the file? I don't think this is really touched often, so I don't expect there to be rebasing problems and we should try to get this into sage-5.12 IMO.

Best,
Travis

[chapoton]

Hello again,

Thanks for looking at my code. Here is an updated patch.

• I have removed the "#indirect doctests" in _div_
• I have added two .parent() tests in _div_
• I have set back the milestone to 5.12 (I have no good reason to postpone)

[tscrim]

Looks good to me. Thanks Frederic.