I don't think we can hope to have a mathematically consistent
definition of numerator and denominator which works consistently in
all rings and fields! I guess that in the cases where we think we
know what we mean, it is because the ring in question is constructed
(mathematically) from another ring via the ring-of-fractions
construction (denoted S^{-1}R in books), but there can be many ways of
getting to the same ring. In the example cited Q[x] can be obtained
from Z[x] by inverting all nonzero constants, which looks fairly
canonical until you think of it as a subring of the field Q(x) which
is its field of fractions...
Hence there will never be a "right answer" to this which happens
automatically. Instead, where there is a useful definition we can
implement it on a case-by-case basis with clearly written docstrings
and examples to show what the functions do and do not do.
Having said that it would be good to have consistent behaviour between
Q[x] and Q[x,y].
John
2009/7/10 luisfe <lftab...@yahoo.es>:
>
>
>
> On 9 jul, 16:36, William Stein <wst...@gmail.com> wrote:
>> On Thu, Jul 9, 2009 at 1:34 AM, John Cremona<john.crem...@gmail.com> wrote:
>>
>> > I can see another problem here. One could argue that the
>> > denominator of (x+y)/2 was 2 while you code gives 1. So the
>> > documentation needs to be very clear, stating that the denominator of
>> > ever polynomial in any number of variables over a field is always 1.
>>
>> > John
>>
>> John raises an extremely important and subtle point. This is a
>> *MAJOR* problem since currently we have
>> {{{
>> sage: R.<x> = QQ[]
>> sage: denominator(x/2)
>> 2}}}
>>
>> however you're proposing making it so
>> {{{
>> sage: R.<x,y> = QQ[]
>> sage: denominator(x/2)
>> 1
>>
>> }}}
>>
>> That's very worrisome, simply because it is inconsistent. Either you
>> change the existing code for univariate polynomials, hence break
>> backwards compatible that can possibly result in subtle errors all
>> over the place (in the sage library and in random user code "out
>> there"), or you change the semantics of denominator for multivariate
>> polynomials to be consistent with the semantics for univariate
>> polynomials.
>>
>> For the record, Magma has a Denominator function that is defined for
>> about 15-20 different types, but does not apply to multivariate
>> polynomials.
>>
>
> Good point. The thing is not clear:
>
> {{{
> sage: R.<x>=QQ['x']
> sage: denominator(x/2)
> 2
> sage: denominator(R.fraction_field()(x/2))
> 1
> }}}
>
> Should it be the same result for univariate polynomials and the
> coercion to univariate rational functions?
>
> What are the reasons to implement denominator this way for univariate
> polynomials?
>
> I have to take a look to the logic behind this, there are further
> problems (maybe another track)
>
> {{{
> sage: N.<a>=NumberField(x**2-5/2)
> sage: denominator(1/a)
> 5
> sage: numerator(1/a)
> ---------------------------------------------------------------------------
> AttributeError Traceback (most recent call
> last)
>
> /home/luisfe/.sage/temp/mychabol/4554/
> _home_luisfe__sage_init_sage_0.py in <module>()
>
> /opt/SAGE/sage/local/lib/python2.5/site-packages/sage/misc/
> functional.pyc in numerator(x)
> 686 if isinstance(x, (int, long)):
> 687 return x
> --> 688 return x.numerator()
> 689
> 690 def numerical_approx(x, prec=None, digits=None):
>
> AttributeError: 'sage.rings.number_field.number_field_element_quadr'
> object has no attribute 'numerator'
> }}}
>
> Even not every univariate polynomial ring has defined numerator
>
> {{{
> sage: R.<y>=N[]
> sage: denominator(y)
> 1
> sage: numerator(y)
> ---------------------------------------------------------------------------
> AttributeError Traceback (most recent call
> last)
>
> /home/luisfe/.sage/temp/mychabol/4554/
> _home_luisfe__sage_init_sage_0.py in <module>()
>
> /opt/SAGE/sage/local/lib/python2.5/site-packages/sage/misc/
> functional.pyc in numerator(x)
> 686 if isinstance(x, (int, long)):
> 687 return x
> --> 688 return x.numerator()
> 689
> 690 def numerical_approx(x, prec=None, digits=None):
>
> AttributeError: 'Polynomial_generic_dense_field' object has no
> attribute 'numerator'
> }}}
> >
>
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