Nils Bruin wrote:
> -1. While I agree that defaulting to matrices over QQ rather than over
> ZZ would lead to more expected behaviour for most users, I don't see
> how the rule for changing the base ring can be made both consistent
> and cheap.
>
> Imagine R1 = QQ[x,y]/(x^2+y^2-1). Then FieldOfFractions(R1) is well-
> defined, so one would expect that
> matrix([x]) would be over the field of fractions.
>
> If R2=Q[x,y]/( x^2-y^2), FieldOfFractions(R2) does not exist. Deciding
> whether FieldOfFractions(Ri) exists is a fairly expensive operation in
> these cases.
Good points. In the current case, I don't think the speed issues affect
us because apparently sage can't create the field of fractions of the
first case, unless I'm doing this wrong:
sage: R.<x,y>=QQ['x','y']
sage: Frac(R.quo(x^2+y^2-1))
---------------------------------------------------------------------------
<type 'exceptions.NotImplementedError'> Traceback (most recent call last)
/home/grout/<ipython console> in <module>()
/home/grout/sage/local/lib/python2.5/site-packages/sage/rings/fraction_field.py
in FractionField(R, names)
104 if not ring.is_Ring(R):
105 raise TypeError, "R must be a ring"
--> 106 if not R.is_integral_domain():
107 raise TypeError, "R must be an integral domain."
108 return R.fraction_field()
/home/grout/sage/local/lib/python2.5/site-packages/sage/rings/quotient_ring.py
in is_integral_domain(self)
351
352 """
--> 353 return self.defining_ideal().is_prime()
354
355 def cover_ring(self):
/home/grout/sage/local/lib/python2.5/site-packages/sage/rings/ideal.py
in is_prime(self)
402 NotImplementedError
403 """
--> 404 raise NotImplementedError
405
406 def is_principal(self):
<type 'exceptions.NotImplementedError'>:
sage: Frac(R.quo(x^2-y^2))
---------------------------------------------------------------------------
<type 'exceptions.NotImplementedError'> Traceback (most recent call last)
/home/grout/<ipython console> in <module>()
/home/grout/sage/local/lib/python2.5/site-packages/sage/rings/fraction_field.py
in FractionField(R, names)
104 if not ring.is_Ring(R):
105 raise TypeError, "R must be a ring"
--> 106 if not R.is_integral_domain():
107 raise TypeError, "R must be an integral domain."
108 return R.fraction_field()
/home/grout/sage/local/lib/python2.5/site-packages/sage/rings/quotient_ring.py
in is_integral_domain(self)
351
352 """
--> 353 return self.defining_ideal().is_prime()
354
355 def cover_ring(self):
/home/grout/sage/local/lib/python2.5/site-packages/sage/rings/ideal.py
in is_prime(self)
402 NotImplementedError
403 """
--> 404 raise NotImplementedError
405
406 def is_principal(self):
<type 'exceptions.NotImplementedError'>:
Of course, we shouldn't rely on Sage not having functionality
(especially if we point it out the lack of said functionality, as it is
quite likely to be fixed soon!).
If it is too expensive in general, then maybe we can just special-case
the ZZ->QQ conversion. My guess is that this would take care of most
common scenarios.
Is that special case what you are referring to below?
>
> I think special-casing matrix([ZZ(1)]) etc. is a bad idea. You would
> need a method R.has_an_obvious_field_of_fractions() on rings to both
> deal with ZZ and avoid expensive operations for R1 and R2 above.
> --~--~---------~--~----~------------~-------~--~----~
Thanks,
Jason
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