Hi!
On Nov 6, 6:10 am, Nick Alexander <[EMAIL PROTECTED]> wrote:
> >> Would you consider this weird if you read it in a paper, or
> >> would you know how to interpret it?
>
> >> "Let $f = x^3 + x + 1$ and consider $f(10)$."
>
> > I'm not so sure I know what to do with that.
Neither am I.
<pedantic>
If I read $f(x) = x^3 + x + 1$ in a paper then I would interpret f as
a polynomial function and hence expect $f(10)=10^3+10+1$.
But in the way William stated it, I would interpret f as a polynomial,
not a function. And then, being referee for Williams paper and being
in pedantic mode, I might complain that he should use $ev(f,10)$
(evaluation) or something like that, since polynomials aren't
functions.
>From that point of view, I think that Sage should distinguish
polynomials and polynomial functions. Of course, there should be an
easy transition between the two things, say, if f is a polynomial in x
and y, then f.as_function() should be a function in two variables. And
there should be an evaluation function, say, f.eval(10,15), which does
the same as f.as_function()(10,15)
</pedantic>
However, as a nice person and reader of Williams paper, I would
probably correctly guess that $f = x^3 + x + 1$ denotes a polynomial
*and* a polynomial function.
Moreover, if we talk about *commutative* polynomials then the
interpretation x^3(10) = x(x(x(10))) suggested by Robert simply can't
be the one we want, since then x*y(10) = x(y(10)) \not= y(x(10)) =
y*x(10).
Hence, I think it is ok that multivariate commutative polynomials in
Sage are callable, by being considered as polynomial functions.
But for (freely) non-commutative polynomials I agree with Robert's
objection. If f is a non-commutative polynomial in two variables, then
I would want that f(a,b) is only defined for *callable* objects a and
b, and would return a callable object, by interpreting x*y(a,b) =
a(b(.)) and y*x(a,b) = b(a(.))
Cheers
Simon
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