[Axiom-developer] [#84 conjugate(c)] [#84 conjugate(c)] (new)

[Bertfried Fauser]

On Fri, 28 Jan 2005, Tim Daly wrote:

Hi,

perhaps I miss the point, but I want to comment a few statements:

> > The problem, as I see it, is that there are subtle degrees of meaning
> > that are easily stepped around when you work on paper but must be
> > clarified in computational mathematics.

I wouldn't even call this subtle, but a core problem.

> Why cannot we allow *all*?  I expect I should be able to coerce 'A+1' to
> an Integer, Polynomial Integer, Expression Integer, etc.  The default
> seems to be Integer, which seems fine.

Why I would have no problem to coerce 'A' to *all* (anything), I see a
problem with the '1', since eg a Ring may not have a '1' and '1' may not
even mean 'unit' (of multiplication, eg an additive group)

> I'm surprised this doesn't work.  A Polynomial on a Ring is still a
> member of that ring and should inherit its functions.

A polynomial on a Ring is _not_ an element of the Ring! The *value* of a
polynomial on a Ring element is a ring element (not even its roots, and
not even limits of polynomials in the dgree may have this property). A
polynomial lives (relatively natural) in the product ring RxRxR...
of the ring R, where addition, multiplication etc are inherited
canonically. However, one can make RxRx... in quite different ways into a
Ring (and thats needed eg in topology) so AXIOM will need to know what one
assumes about this extension.

> (in this case,
> conjugate)  It also strikes me that Polynomial Complex Integer is the
> proper type here, not Complex Polynomial Integer...clearly they are
> inequivalent.

You shoul be aware that the complex numbers, seen as a field, and the ring
C, seen as R-algebra (R real numbers, eg. pairs of real numbers) are *not*
isomorphic objects w.r.t. complex conjugation. Only the R-algebra C has
the obvious complex conjugation, while C as a filed has infinitely many
inequivalent (non constructive) conmplex conjugations. Hence AXIOM is
right to be careful here.

Is this of any help?
ciao
BF.


% PD Dr Bertfried Fauser
%     Institution: Max Planck Institut for Math, Leipzig <http://www.mis.mpg.de>
%   Privat Docent: University of Konstanz, Phys Dept 
<http://www.uni-konstanz.de>
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%          Phone : Leipzig +49 341 9959 735  Konstanz +49 7531 693491

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