solve remembers too much, asks odd questions.

On 11/7/2010 2:29 PM, Michel Talon wrote:
> Richard Fateman wrote:
>
>
>
>> A solution is discussed here ..
>> http://www.cs.berkeley.edu/~fateman/papers/y=z2w.pdf
> I have read the paper and i do not agree with what you are saying.
> sqrt(z) has 2 values for any complex values of z, except 0 and
> infinity.

I am confused by your statement.  I glanced through the paper
and I notice only one mention of a value for sqrt(r), where it
says the positive sqrt(r) of the positive real number r.
Finding solutions for y=z^w is the issue.
>   This shows the inanity of the conventional cut on the negative
> real axis which indeed leads to the difficulties you mention.

This is, as you say, the conventional place for the branch cut.  This is 
not the
cause of problems, because any other location would also be arbitrary.
> In fact
> the cut between 0 and infinity is completely arbitrary and can be
> any curve in the complex plane joining these two points, which are
> themselves intrinsic as branch points of a branched covering.
>
> The real problem is not this one, it is "how many values take
> sqrt(2) + sqrt(5)" for example, or worse "sqrt(2)+5^{1/3}".
I would state it differently.  What does the notation sqrt(2)+sqrt(5)  
mean when you write it down,
or print it out,  in YOUR computation?

> Or for another
> example, how many values take the Cardan solutions of the degree 3
> equations. You know in this case there are 3 solutions, which is not
> immediately obvious from the Cardan formulas.
Not only that, substitution into the formulas in a naive manner 
generally does not work.

> The answer to such questions lies in the theory of Riemann surfaces, and
> requires non trivial analysis of the considered algebraic equation. One may
> represent the Riemann surface as a branched covering of some degree, with
> some arbitrary well chosen cuts between the branch points. Then one may
> define a determination of the algebraic function on the cut complex plane.
Is there some part of this that cannot be computed by a computer but can 
be computed by
a human? (in principle).  If so, identifying it would be most valuable, 
for reasons that
range from philosophical to theological.

If these parts can be computed, it would be nice to describe how, so 
someone can program them.
(See work by Adam Dingle, for a partial solution.)


>> or just letting the program run and see what comes out.
> Still this is the main benefit one can get from a CAS program in case it
> finds a non obvious simplification.
There are non obvious and incorrect simplifications.    How do you avoid 
those??

>   There is nothing as irritating as a CAS
> which is unable to simplify an expression with a bunch of square roots
> without getting "expression too large errors" (i got that under Maple)
That was probably a long time ago. Maple can handle much larger 
expressions now.
>   and
> wasting hours of your time for computations that you finally do by hand in a
> couple of minutes. All these sign considerations and "assumptions" get in
> the way of the computation, for zero benefit in general. Expecting a CAS to
> give correct results for non trivial computations (integrals, etc.) is not
> reasonable, one certainly needs to check the result by all sorts of means,
I disagree, in the sense that the goal of (most) CAS developers is 
exactly to give correct results.
>
> but a CAS may be of great help by finding an unexpected  "simple" form.
> And of course one expects a CAS to give correct results for "trivial"
> computations, that is purely polynomial computations, however large they may
> be.
>
>
>