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MathGroup Archive 2004

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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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