[FOM] When is it appropriate to treat isomorphism as identity?

[Andrej Bauer]

On Wed, May 20, 2009 at 7:58 PM, Jay Sulzberger <jays at panix.com> wrote:
> Given a polynomial in one variable of degree 10 with rational
> coefficients, which can be written out in the usual high school
> notation with less than 1000 characters, can you compute the
> "number of distinct roots"?  The polynomial is to be given as an
> ascii string placed before you, so you can see it.

For _rational_ coefficients, the mapping from polynomials to the
number of distinct roots is computable. For real coefficients it is
not.

I am not talking about feasible computation, but about computability.

> Free example:
>
> x^10 + (7777777777777/666666666666666)x^5 + (-1111111111111)x^4 + (17)x^0

This particular polynomial has 4 distinct real roots and additional 6
distinct complex roots. In this age of Mathematica and fast computers
you'll have to pick nastier coefficients to get Wolfram embarrassed.

But what point are you trying to make?

With kind regards,

Andrej Bauer