Hi Andrea,
On 2013-04-17, Andrea Lazzarotto <andrea.lazzaro...@gmail.com> wrote:
>> Note that by saying
>> S(x) = ...
>> you define S as a symbolic function on a symbolic variable x, and if you
>> re-define x later, then the variable of S will still be symbolic, and
>> not belong to a polynomial ring.
>>
>
> Ok, my bad. I am still fairly new to Sage (even if I tried to submit a
> super small patch) and I don't completely understand what's the difference
> here,
A symbolic expression is something fairly general---so general that many
methods won't make sense in this generality. A symbolic expression may
contain all "strange" stuff like sin(x^2), exp(pi), x^x, etc.
In contrast, a polynomial ring is a structure that has a fixed base ring
(finite field of order 16, in your example) and has one or several
generators (x, in your example), so that *all* elements are sums of
terms of the form "element of the base ring times product of
generators". Hence, x^x is not a polynomial (the exponent must be a
natural number), there is no sin, no pi, etc.
Polynomials are of course quite useful in some branches of mathematics.
And concerning your problem: Polynomial rings over fields are Euclidean
domains, and thus there is an algorithm to compute quotient with
remainder or gcd and so on.
>From computer science point of view, the mathematical algorithms are
implemented in methods of certain (base) classes. The polynomials we want
to consider here belong to classes that provide a quo_rem method. But
general symbolic expressions belong to a different class, and a quo_rem
method would make no sense for them.
In your example, a particular situation occurs: You define a and b.
While a *is* a polynomial, b is not. Hence, the method a.quo_rem
exists, but the method realises that b is no suitable input.
> Unfortunately I am an undergraduate student in Computer Science (not even
> math) so there are some concepts which are still obscure to me. :P For
> example, what is the meaning of these two lines?
>
> P.<x> = f[]
This line is a shortcut for
P = PolynomialRing(f,'x')
x = P.gen()
It is syntactical sugar: P.<x>=f[] is not valid in Python, but it is a
convenient notation well known from the Magma computer algebra system.
Hence, the Sage developers wanted to make this notation available, by
means of a pre-processor.
> x = P.gen()
As I mentioned above, a polynomial ring knows its base ring and it
generator(s). Here, we have the base ring f, which you could check by
sage: P.base_ring() is f
True
And P.gen() returns the generator (i.e., the "variable") of this
polynomial ring. Note that this "variable" is by no means the same as a
symbolic variable of the same name:
sage: x = var('x')
sage: x
x
sage: P.gen()
x
sage: type(x)
sage.symbolic.expression.Expression
sage: type(P.gen())
sage.rings.polynomial.polynomial_zz_pex.Polynomial_ZZ_pEX
sage: hasattr(x, 'quo_rem')
False
sage: hasattr(P.gen(), 'quo_rem')
True
By the way, you can get the documentation of a method by appending a
question mark (and hitting return resp. shift-return in the notebook).
Hence, P.gen?<ret>
By appending two question marks, you can get the source code (in most
cases). If you start typing the name of a method and hit the tab key,
you'll get a list of all methods starting with what you typed. Hence,
P.ge<tab> will give you P.gen, P.gens, P.gens_dict, P.gens_dict_recursive,
P.get_action, P.get_action_c, P.get_action_impl, for each of which you
could then check the documentation.
Best regards,
Simon
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