I would think that the method `__hash__` of FractionFieldElement in
fraction_field_element.pyx is broken, since
sage: f1 = x/y
sage: f2 = (2*x)/(2*y)
sage: f1 == f2
True
sage: hash(f1)
-284264079394034550
sage: hash(f2)
-284264773958195866
In `__hash__`, we do the following:
if self._parent.is_exact():
...
self.reduce()
# Same algorithm as for elements of QQ
n = hash(self._numerator)
d = hash(self._denominator)
if d == 1:
return n
else:
return n ^ d
The problem is that `self.reduce()` doesn't have any effect: it divides out
the gcd of numerator and denominator, which is 1, since QQ is a field.
(Over ZZ it is 2, which is the reason why it works)
I don't know how to fix this properly. Can we define a sensible hash
generically for FractionFieldElement at all?
Partially, this might also be my fault from
https://github.com/sagemath/sage/pull/38924
There are other tickets that thought about this, e.g.,
https://github.com/sagemath/sage/issues/16268
:-(
Martin
On Wednesday, 16 April 2025 at 17:48:47 UTC+2 Nils Bruin wrote:
> On Wednesday, 16 April 2025 at 04:55:39 UTC-7 Peter Mueller wrote:
>
> The following code
>
> K.<x, y> = QQ[]
> K = K.fraction_field()
> print(len({x/y, (2*x)/(2*y)}))
>
> gives the answer 2, even though the two elements of course are the same!
> Is this a bug or a feature for a reason I cannot guess? Same on the
> SageMath Cell.
>
>
> I don't think it's a feature but it might be that you're hitting general
> code that can't do much more than it does. In that case we should probably
> have a specialization that deals with that particular situation.
>
> In your case, we can just force the denominator to be monic. It can make
> for less nice representations because it might cause fractional
> coefficients in the numerator and denominator:
>
>
> f.numerator()/(c:=f.denominator().leading_coefficient())/(f.denominator()/c)
>
> For this standardization, we need that there's a monomial ordering (which
> would generally be met) and that the leading coefficient is a unit (true
> over a field). It's the last one that is generally problematic. In
> ZZ["x,y"].fraction_field(), you'd already need a different approach and
> over domains with more complicated unit groups and/or without unique
> factorization, normalizing the denominator is going to be very expensive.
> Note that it doesn't affect the ability to compute in the field of
> fractions: equality testing is still easy. It's just the normal form that's
> hard (and which is necessary to get to a well-defined hash).
>
> Funniliy enough:
>
> K.<x, y> = ZZ[]
>
> K = K.fraction_field()
> print(len({x/y, (2*x)/(2*y)}))
>
> so it seems that the extra work was already done in that case. And that's
> also the representation in which you'll avoid denominators in the
> denominator! So probably it's better to switch to that representation. If
> you need polynomials you can use
>
> QQ['x,y'](f)
>
> when the denominator has degree 0.
>
>
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