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

From: Ralf Hemmecke
Subject: Re: [Axiom-developer] Am I posing this solve problem wrong?
Date: Mon, 05 May 2008 09:43:39 +0200
User-agent: Thunderbird (X11/20080421)

On 05/05/2008 05:46 AM, Bill Page wrote:
On Sun, May 4, 2008 at 10:23 PM, C Y wrote:
--- Bill Page <address@hidden> wrote:

I guess you need to be a little more explicit about the variables:
Ah.  Thanks Bill.  I wonder if this is the desired default behavior?

Cliff, I guess, you asked for the result you got. Bill has just shown you how to get what you actually had in mind.

Well, Axiom is a strongly typed system so specifying the main
variables of a multivariate polynomial with coefficients from some
other ring is not entirely unexpected.

Right. Now look at Cliff's original expression.

(1) -> E1 := x2*D12+(y-y1)2*C12 - C12*D12
(1) ->
         2  2      2         2 2     2 2     2  2
  (1)  C1 y1  - 2C1 y y1 + C1 y  + D1 x  - C1 D1
                         Type: Polynomial Integer

Without further information, Axiom puts it into the domain Polynomial(Integer). *Don't* think that this type represents a polynomial in x and y with C1 and D1 being in the coefficient domain.
None of these variables is a distinguised one. If you now say


then before looking at the result, think of what would *you* return.
Maybe you are asking for an expression for y. But, hey, that is of course not the only thing that you could have asked for. 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


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.

Now, if Axiom had given you

               +----------+                 +----------+
               |   2     2                  |   2     2
            D1\|- x  + C1   + C1 y1    - D1\|- x  + C1   + C1 y1
   (2)  [y= -----------------------,y= -------------------------]
                       C1                          C1
                                Type: List Equation Expression Integer

that would even be wrong, since your intention was to get elements of R.
Having types, Axiom should actually allow you to specify what you want.

  E1: P := x^2*D1^2+(y-y1)^2*C1^2 - C1^2*D1^2

Unfortunately, that doesn't work (yet), but I would expect it to work in an ideal AXIOM.


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