       Re: Re: Please, can someone explain this small function?

• To: mathgroup at smc.vnet.net
• Subject: [mg50833] Re: [mg50808] Re: Please, can someone explain this small function?
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Thu, 23 Sep 2004 05:27:14 -0400 (EDT)
• Organization: Mathematics & Statistics, Univ. of Mass./Amherst
• References: <cih0hn\$md\$1@smc.vnet.net> <200409220411.AAA18689@smc.vnet.net>
• Reply-to: murray at math.umass.edu
• Sender: owner-wri-mathgroup at wolfram.com

```One reason to treat a polynomial as a list -- and that might not be the
original poster's reason -- is that one definition of "polynomial" is as
just such a list, the list of its "coefficients" (with some convention
about an ascending or descending order).

The reason for that is the conventional definition of a polynomial as
"an expression of the form c0 + c1 x + c2 x^2 + ... + cn x^n" is a bit
mysterious.  Yes, one can write down such an expression, but just
because one can write it down doesn't necessarily mean it corresponds to
any fundamental mathematical reality.  How can one give meaning to such
an "expression" in terms of more fundamental mathematical entities.
(And here we must be careful to distinguish between a polynomial, which
is one thing, and a polynomial function which is quite another thing --
most particularly when the coefficient domain is not the reals or
complexes.)

It's a similar situation with respect to the definition of complex
numbers.  One can say that a "omplex number" is an "expression of the
form "a + b i" where a and b are real and i is an object satisfying i^2
= -1, but that in itself doesn't really say what such an object actually
is.  Recall the history here: Folks used such expressions a + b i for a
long time but felt very uncomfortable doing so -- until eventually the
definition of a + bi as meaning the ordered pair (a, b) was offered,
which expressed the new kind of object in terms of already understood
objects.

All that said, most folks do find it more convenient to write and
manipulate polynomials in the "expression in x" form.  After all,
traditional mathematical notation -- with all its ambiguities and
limitations -- was devised so as to make it easy to write polynomials!

Paul Abbott wrote:
> In article <cih0hn\$md\$1 at smc.vnet.net>,
>  Cole Turner <REMOVEcole.turner at liwest.at> wrote:
>
>
>>input: two polynomials as lists
>
>
> Why not input polynomials as polynomials? ...

--
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305

```

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