NetNews Usenet Archive 1992 #16/NN_1992_16.iso/spool/sci/math/9350

Internet Message Format | 1992-07-20 | 2.4 KB

Path: sparky!uunet!munnari.oz.au!ariel!ucsvc.ucs.unimelb.edu.au!u7023595
From: [e-mail]
Newsgroups: sci.math
Subject: Re: Roots Of Integer Poly's
Message-ID: <[e-mail]>
Date: 21 Jul 92 07:23:56 GMT
References: <[e-mail]>
Organization: The University of Melbourne
Lines: 45

>In article <[e-mail]>, [e-mail] (David Wilson) writes:
>
>>I wonder if any one out there knows anything on the following.
>>
>>Let F(x) be a monic polynomial with integer coeff's of degree n.
>>Let F(x) have roots   \alpha_{1}, ... , \alpha_{n}  is there a
>>constant C depending only on n such that
>>
>>       | \alpha_{j} - \alpha_{i} | >= C
>>
>>For all i,j. Ie the roots cannot be too close together.
>>This is trivially true for n=1 or n=2 but what about general n ?
>
> [Stuff deleted]
>
> For n > 2, let p(x) = x^n+bx^(n-1)-x^(n-2).  For b >= 3, p has the
> distinct real roots 0 and (sqrt(b^2+4)-b)/2, which differ by less than
> 1/b.  Hence these roots may be brought arbitrarily close together by
> choosing b sufficiently large, and there is no positive lower bound on
> the distance between distinct roots of p.
>
WLOG assume the monic polynomial is irreducible over the rationals as
this is the interesting case.
For n=2 it is easily verified that \sqrt(3) is the lower bound, attained
for x^2 + x + 1 (cube roots of unity).
Consider n > 2.
In Wilson's example (not irreducible) the roots are 0, (-b + \sqrt(b^2+4)/2
and (-b - \sqrt(b^2+4)/2.  The difference between the latter two roots is
\sqrt(b^2+4) > 2 , not less than 1/b as claimed.
Actually the answer to the question posed above is YES.
In 1930 J. Favard showed that one can take C = \sqrt(3/2).
Blanksby, Lloyd-Smith and McAuley gave C = \sqrt(3) + \epsilon for n
sufficiently large and suitable \epsilon > 0.
The problem is completely solved through the efforts of Langevin, Reyssat
and Rhin. They show that C = \sqrt(3) always works. Also they found that
one can take C = 2 - \epsilon for given \epsilon > 0 for n sufficiently
large.
The second edition of Narkiewicz's book on algebraic number theory gives
some references (first chapter? and also a postscript at the back of the
book). Also see M. Langevin, E. Reyssat and G. Rhin "Diametres Transfinis
et Probleme De Favard", Annales Institut Fourier de l'Universite de Grenoble,
1988 and M. Langevin "Solution des problemes de Favard" in same publication
(both Vol. 38).

Bill Lloyd-Smith