I think this is kind of in the right direction, but there are ambiguities
about
polygcd too.
I think that there should be integer gcd and integer extended gcd. But for
polynomials
it matters what the coefficient domain is.
For example, what is the GCD of
(x/4, x^2/4) ? these are not polynomials over the integers, but over
the rationals. GCD might be x, or x/4.
what about (1/(4*x), 1/(8*x)) ? Perhaps 1/(4*x) or 1/(8*x) or 1/x or
maybe
even 1.
It is possible to describe this stuff in substantial details, but would
require
the careful definition of the terms "content", "primitive part", "unit" as
well
as ring and field.
so maybe if we define polygcd as polynomial GCD of polynomials (perhaps
in several variables) over the integers, we are OK. But then trying to
compute
the GCD when the coefficients are NOT integers, but rational numbers or
for that matter, rational functions themselves, could be (a) an error or
(b) something we define in some particular way that might not be correct.
There is a note in sci.math.symbolic that points out, I think, in some
system, that
GCD(i,i) is not i, but 1. i is sqrt(-1). So there is also a GCD over the
"Gaussian Integers"
that may or may not be intended.
RJF
----- Original Message -----
From: "Robert Dodier" <robert.dodier at gmail.com>
To: "Richard Fateman" <fateman at cs.berkeley.edu>; <macrakis at gmail.com>
Cc: "Maxima" <maxima at math.utexas.edu>
Sent: Friday, March 17, 2006 7:17 AM
Subject: Re: [Maxima] gcd strangeness
> hello richard and stavros,
>
> about this proposal,
>
>> (1) we can rename gcd to polygcd and create a new gcd function.
>> The new gcd would treat its arguments as numbers
>> (or expressions which eventually evaluate to numbers).
>
>> About (1), we recently went through a similar discussion
>> about mod and nummod. The outcome was that mod was
>> renamed to polymod and nummod to mod.
>
> i wonder if you are willing to take a stand either for or against
> this proposal.
>
> all the best,
> robert dodier
>
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