On Wed, Jul 9, 2008 at 10:16 PM, Yixin Cao
<[EMAIL PROTECTED]> wrote:
>
>
> On Wed, Jul 9, 2008 at 9:05 PM, Bill Page <[EMAIL PROTECTED]>
> wrote:
>>
>> On Wed, Jul 9, 2008 at 8:43 PM, Gabriel Dos Reis
>> <[EMAIL PROTECTED]> wrote:
>>
>> > ...
>> > In the field of fraction of polynomial, the fact that 1/(2d) is well
>> > defined seals no doubt. It does not matter whether d can take
>> > on the value zero. That is just a simple fact. That is one of
>> > reasons why simplification of expressions that assume
>> > fraction of polynomials prove inadequate some times.
>> >
>>
>> If "d can take on the value zero" doesn't that imply we are thinking
>> of '1/(2d)' as a function?
>>
>> Isn't the following result another example such an error?
>>
>> (1) -> q:=((x^2-1)/(x-1))
>>
>> (1) x + 1
>> Type: Fraction Polynomial
>> Integer
>> (2) -> q(1)
>>
>> (2) 2
>> Type: PositiveInteger
>>
>> Surely the fact that we can write (2) means this simplification is
>> unjustified.
>
> Just like Gaby has pointed out a little earlier, in this example q is a
> fraction polynomial function, instead of fraction polynomial.
No, that is not true. q is a fraction of polynomial. When one writes
q(a) where q is an element of R[X] and a is an element of R, the common
interpretation from most algebra texts is that you consider a ring
morphism
eval_a: R[X] -> R
that sends X to a, and you consider the image of q under eval_a.
q is not interpreted as a function.
You should not be fooled by the syntax.
-- Gaby
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