Re: [Axiom-developer] Am I posing this solve problem wrong?

[Ralf Hemmecke]

On 05/05/2008 04:03 PM, Bill Page wrote:
> On Mon, May 5, 2008 at 3:43 AM, Ralf Hemmecke wrote:
> > ...
> > Having types, Axiom should actually allow you to specify what you want.
> >
> >  R:=MPOLY([C1,D1,y1,x],Integer)
> >  P:=MPOLY([y],R)
> >  E1: P := x^2*D1^2+(y-y1)^2*C1^2 - C1^2*D1^2
> >  solve(E1=0,y)@List(R))
> >
> > Unfortunately, that doesn't work (yet), but I would expect it to work in an
> > ideal AXIOM.
>
> I do not think this can work exactly like this since the solution
> cannot be expressed in R.

I refer again to what I wrote in
http://lists.gnu.org/archive/html/axiom-developer/2008-05/msg00022.html

--BEGIN quote
Suppose, I consider the above input expression as an element in R[y]] 
where the coefficient domain R is Z[C1,D1,y1,x]. And I would like

  solve(E1=0,y)

to stand for "find a (or several) z \in R such that E1[y<--z]=0" where 
E1[y<--z] is my (fancy) notation for substitution of y by z in E1.
--END quote

I have a polynomial E1 in R[y] and I am looking for those z \in R which 
make this (univariate) Polynomial vanish. If there are no such z \in R 
then Axiom should return an empty list (or empty set).

> You need Expression in order to include
> kernels involving 'sqrt'.

I am not looking for the most general expression.

> Or did you have something else in mind?

Yep. I was looking for solutions that live in R.

> Inspite of that, could you explain a little more about how you think
> the ideal AXIOM should work? It seems to me that the way it works now
> is rather complex and is a consequence of several interacting factors
> such a coercion rules that heuristically start with the most specify
> type (i.e. a polynomial type) using coercion to expand the range of
> possibilities.  This is affected in a non-trivial and not so easily
> predicted manner by the exports exports from the underlying types.
> This is built into the Axiom interpreter in a rather deep way. Is
> there a different way this could work that might produce less surprise
> (and frustration) on the part of the user?

Ideally, I would like to tell "solve" where it should look for 
solutions. Consider the simple equation.

  x + 1 = 0

What does "solve for x" actually mean? It is ambiguous.
I would like to say

  solve(x+1=0, x)$Set(PositiveInteger)
  solve(x+1=0, x)$Set(Integer)

(or similar) and get different results.

Of course, now one could say, that one first computes the most general 
expression for the result. And then filters out results that one is not 
interested in.

So solve polynomials over the complex numbers and filter out all real 
solutions, if you are only interested in real solutions.
That's a simple case.

But I guess, there are cases where one would simply switch to another 
(maybe faster) algorithm if it were known at input time that some of the 
results are not needed anyway. Why should one bother to waste 
computation time if the result of that computation is thrown away 
immediately afterwards? Imagine one has given a polynomial (of high 
degree) with integer coefficients and looks only for integer solutions?
Root isolation can stop if the interval is less than 1 since either 
there is exactly one solution or none in this interval. (Maybe not the 
best example to demonstrate what I mean.)

Ralf