On 15 May 2010 22:09, Robert Bradshaw <rober...@math.washington.edu> wrote:
> On May 15, 2010, at 8:54 AM, Simon King wrote:
>
>> Hi John!
>>
>> On 15 Mai, 17:34, John Cremona <john.crem...@gmail.com> wrote:
>>>
>>> ...
>>> There's a more general issue here, perhaps. In your R = Z[[x]], you
>>> ask for the inverse of x, which is not invertible as an element of R.
>>> The conservative response is to return 1/x in the smallest ring
>>> containing R in which x has in inverse, which is Laurent series over
>>> Z. The simpler response is to say "since R is an integral domain,
>>> let's form its field of fractions and do the inversion there".
>>
>> I think there is already a convention in Sage: a/b should *always*
>> live in the fraction field. Therefore, the parent of 1/1 is QQ, not
>> ZZ.
>
> Hmm... I've always found it handy that
>
> sage: R.<x> = QQ[[]]
> sage: 1/(1+x)
> 1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^10 - x^11 + x^12 -
> x^13 + x^14 - x^15 + x^16 - x^17 + x^18 - x^19 + O(x^20)
> sage: parent(1/(1+x))
> Power Series Ring in x over Rational Field
Is this a relevant example? 1+x is a unit in QQ[[]] and no fraction
fields are required.
I think you meant
sage: parent(1/x)
Laurent Series Ring in x over Rational Field
-- but again, this *is* the field of fractions of QQ[[]], so again
there is no inconsistency.
So perhaps you meant
sage: R.<x> = ZZ[[]]
sage: parent(1/x)
Laurent Series Ring in x over Integer Ring
where 1/x does *not* now belong to the fraction field.
John
John
>
> but I guess consistency would be nicer. I'm a bit hesitant to see this
> change go in without some speed regression testing, particularly in some of
> the Monsky-Washnitzer code.
>
> sage: R.<x> = QQ[[]]
> sage: timeit("x*x")
> 625 loops, best of 3: 78.2 µs per loop
> sage: f = 1/x
> sage: timeit("f*f")
> 625 loops, best of 3: 190 µs per loop
>
> - Robert
>
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