On Fri, Aug 22, 2014 at 7:38 PM, rjf <fate...@gmail.com> wrote:
>
> On Friday, August 22, 2014 2:04:35 PM UTC-7, Bill Hart wrote:
>>
>>
>>> RJF said...
>>> I don't know about canonical maps. The term "canonical representation"
>>> makes sense to me.
>>
>>
>> He means this. In algebra Z/nZ is actually a ring modulo an ideal. Z is
>> the ring, nZ is the ideal.
>>
>> The elements of Z/nZ are usually written a + nZ. It means precisely this:
>>
>> a + nZ = { x in Z such that x = a +nk for some k in Z }
>>
>> So it is a *set* of numbers (actually, in algebra it is called a coset).
>>
>> When you write Mod(a, n) you really mean a + nZ if you are an algebraist,
>> i.e. the *set* of numbers congruent to a modulo n.
>>
>> So Z/nZ consists of (co)sets of the form C = Mod(a, n) = a + nZ. If you
>> like Z/nZ is a set of sets (actually a group of cosets in algebra).
>>
>> There is a canonical map from Z to Z/nZ: we map b in Z to the (co)set C in
>> Z/nZ that contains b.
>>
>> If a = b mod n then C = mod(a, n) is the *only* one of these sets which
>> contains b. So the map is canonical. We only have one choice.
>>
>> But there is no canonical map going the other way. For any such (co)set C,
>> which element of Z are you going to pick? Any of the elements of C will do.
>>
>> You can make an arbitrary choice, e.g. pick the unique element of C in the
>> range [0, n). Or you could pick the unique element in the range (-n/2, n/2].
>> But that is not a canonical choice. You had to make an arbitrary choice.
>> Someone else may have made a different choice. (Indeed some people use the
>> latter choice because some algorithms, such as gcd and Jacobi symbols run
>> faster with this choice.)
>>
>> So canonical map means there is only one way you could define the map. You
>> don't need to tell anyone what your map is, because they *have* to make the
>> same choice as you, as there are no alternatives. That's the meaning of
>> canonical in mathematics.
>>
>> It's probably not a terminology they teach in Computer Algebra,
>
>
> There are few courses in Computer Algebra. But the term canonical is used
> differently. In fact, two systems could
> have different canonical forms (e.g. in your example, [0,n-1] or
> [-(n-1)/2, (n-1)/2] for representatives of the numbers mod n.
>
> Just as one system could have 1+x+x^2 and another x^2+x+1.
>
> But for each system with a canonical form for polynomials, all polynomials
> in its canonical form would be
> identical bit patterns. Some systems e.g. Maple go further and have all
> canonical and equal forms stored
> at the same machine address.
I think this highlights the difference between your perspective and
mine. When I write "the map Z -> Z/nZ in Sage" you seem to be
interpreting that as "a map from representatives of the integers to
representatives of the integers mod n" and I'm thinking "a
representation of the (canonical) map Z -> Z/nZ." Same for Rings,
Fields, etc.
>> but it is taught to undergraduates around the world in pure mathematics.
>
> Um, some English speaking students who take, um, certain electives?
Many more than take courses in computer algebra. I'd say.
>> The whole argument here is that because only one direction gives a
>> canonical map, coercion must only proceed in that direction. Otherwise your
>> computer algebra system is making a choice that someone else's computer
>> algebra system does not.
>
>
> But that is perfectly all right. canoncality is with respect to a
> particular computer system. At least
> my kind of canonicality.
Clearly, despite some overlap, you're a computer scientist, and I'm a
mathematician:
http://en.wikipedia.org/wiki/Canonical
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