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Re: [Axiom-developer] about Expression Integer

From: Martin Rubey
Subject: Re: [Axiom-developer] about Expression Integer
Date: 22 Feb 2006 14:54:55 +0100
User-agent: Gnus/5.09 (Gnus v5.9.0) Emacs/21.4

"Bill Page" <address@hidden> writes:

> Martin,
> On February 22, 2006 3:54 AM you wrote:
> > 
> > I'm currently out of time. But I'm a little afraid of the 
> > direction the discussion takes currently.
> I on the other hand am very happy that this discussion
> seems to accurately reflect the concept of type in Axiom.

well, I'd be happier if I could defend my point of view. Just, I can't,
currently. Still I'll try to (partially) respond to this message... Please bear
with me.

> > I believe that the intension of a constructor like DMP is to cover
> > *polynomials*.
> I agree. "cover" is the operative word. This does not preclude that the type
> DMP in Axiom might be a little more general than what is called a polynomial
> in some mathematical uses (but perhaps not others).

> > If we allow (2*x+1/x)::DMP([x], EXPR INT), i.e., allowing 1/x as a
> > coefficient, then it must be clear that the "x" in 1/x is something
> > different than the "x" in 2*x. Otherwise we don't get a polynomial,
> > obviously.
> I think this is wrong. In fact, it is clear from the implementation of Axiom
> that whether the 'x' in these two cases is the same or not depends on which
> domain to which they belong.
> 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 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. 

I guess you know that, so there is probably a misunderstanding somewhere. Just
to be clear:

sin x + y*cos x + y^2* tan x 

is perfectly allright a polynomial in y.

> Documentation is a good thing. But why should we retain behaviour in Axiom
> that is mathematically incorrect? What advantage does this odd behaviour of
> 'differentiate' have over a mathematically correct implementation?

I think that 'differentiate' does not exhibit odd behaviour. Why do you think
it's odd?

Of course, if you claim that x+1/sin(x) is a polynomial, than I'm out of luck.

You said, given

> > > > P := DMP([x,y], EXPR INT) 
> > > > a :P := x 
> > > > b := a/x 
> > > > 
> > > > differentiate(b,x)                    -- 1/x
> > > 
> > > (4) -> differentiate(b,x)
> > > 
> > >         1
> > >    (4)  -
> > >         x
> > >    Type: DistributedMultivariatePolynomial([x,y],Expression Integer)

> On the other hand a 'differentiate' operator that returns 1/x in (4) above
> would violate the mathematical definition of derivative, so I think it cannot
> possibly be right.

I wouldn't say that this violates the "mathematical definition of
derivative". The derivative of a univariate polynomial is defined by sending
x^n to n*x^(n-1) and linear extension. There is no such thing as 1/x in the
polynomial world. If you want a domain that contains x and 1/x, consider using
Laurent polynomials. In fact I cannot see a single reason to consider a domain
DMP(vars, EXPR INT) at all, except for the fact that we do not have a domain
EXPR(vars, coefficients) yet.

> > Note that b::UP([x], INT) or b::POLY INT give errors!
> Obviouse 'b' has no representation in these domains. 

Yes it has: 1 is a UP([x], INT).


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