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Re: [Axiom-developer] about Expression Integer

From: Francois Maltey
Subject: Re: [Axiom-developer] about Expression Integer
Date: 22 Feb 2006 19:11:12 +0100
User-agent: Gnus/5.09 (Gnus v5.9.0) Emacs/21.4

"Bill Page" writes :

> > > Could you please define in what sense "Otherwise we don't 
> > > get a polynomial, obviously."? To me this is not obvious -
> > > it is wrong.
> > 
> > Why should this be wrong?
> > 
> > Here's a definition for polynomial from wikipedia:
> > 
> >   In mathematics, a polynomial is an expression in which constants
> >   and variables are combined using (only) addition, subtraction,
> >   and multiplication. Thus, 7x^2+4x-5 is a polynomial; 2/x is not.

I have the same point of view than Martin, 
everything is done in DMP in order to have a polynomial ring, 
DMP isn't only a << collect >> command which factorise x and so.

> I think this definition is less general it needs to be and
> difficult to apply in the context of Axiom. 

If we change DMP assumption, it's no more ring. 

> We need to know what are "constants" and what are "variables" 

So the same variable is forbiden in the constant ring.
I find it's a bug to allow DMP ([X,Z], DMP ([X,Y], INT))

> > I recall that in the Algebra course I attended, the polynomial
> > ring was defined as a ring (of coefficients) together with a
> > variable which is to be transcendent over that ring. 
> That definition is much better. In the case we are discusing
> the ring of coefficients is the domain 'EXPR INT' which
> includes expressions of the form '1/x' and so it is correct
> to say that '2/x' is not in itself a polynomial but it can
> be a coefficient in a polynomial. 

I don't think so, 
a polynomial ring has variables which are not in the initial ring.
The DMP error (for derivative) isn't in the derivative function but
in the fact that axiom accepts 1/x has variable in DMP[x].

And 1/x isn't transcendent over that rign because y=1/x is solution of
xy-1=0 in this field.

Either we say to the user he _must_ be carreful with this, 
   (only in the EXPR INTEGER field?)
Either monomial command of DMP reject 1/x coefficient if x is a variable.

In France we say we can be << jesuite >> :
The real axiom accepts coefficients as 1/x in DMP ([x], EXPR INT) 
because axiom EXPR INT isn't perfect, but it souldn't.
But I'm sure it's a mistake to change derivative.

> How would you use such a domain to solve the problem originally
> posed by Francois about expansion of trigonometric expressions
> that started this thread? 

When we expand a trigonometric expression, 
the question is a << linear one >> not a polynomial one.

We expand more often sin (4*x+5*y) than sin (x^2+x) ; 
for sin (x^2+x) we use series.

Have a good day !


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