NetNews Usenet Archive 1992 #26/NN_1992_26.iso/spool/sci/math/14798

Internet Message Format | 1992-11-11 | 1.9 KB

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From: [e-mail] (Ronald Bruck)
Newsgroups: sci.math
Subject: Re: symbolic derivation of polynomial roots
Date: 11 Nov 1992 16:55:04 -0800
Organization: University of Southern California, Los Angeles, CA
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In article <[e-mail]> [e-mail] (Alexander Hanysz) writes:
>In article <[e-mail]> [e-mail] (John McKay) writes:
>>
>>My understanding is that the roots of ANY polynomial in Q[x]
>>can be expressed in terms of radicals. What is the trick?
>>You may need infinitely many of them.
>
>Well, if you're going to be like that then you don't even need radicals!
>Any real number can be expressed in terms of integers and arithmetic
>operations.  You just need infinitely many of them.  It does seem to
>trivialise the issue somewhat.
>
>                                Alex.
>

I think this misses the point.  As I understand the question (although
I don't know the answer), he's asking whether there is some closed-form,
though possibly infinite in extent, formulation of the roots in terms of
the coefficients of the polynomial.  For example, it might mix continued
fractions with expressions like $\sqrt{a_0 + \sqrt{a_1 + \dots$, where
the $a_i$ have explicit expressions in terms of the coefficients.  It's
this dependence on the coefficients which makes the example of "every real
has an expansion into an infinite decimal" spurious.

He's asking a serious question, and if his formulation of it is a little
vague, ***that is the nature of all research problems***, until the
canonizers have reduced it to Definition 2.1.7 of Section 3.2, after which
I cease to be interested.

I'm curious as to the answer.  Is there an interesting formulation?

--Ron Bruck
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