NetNews Usenet Archive 1992 #16/NN_1992_16.iso/spool/sci/math/9429

Internet Message Format | 1992-07-22 | 1.4 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: 23 Jul 92 01:38:37 GMT
References: <[e-mail]> <[e-mail]>
Organization: The University of Melbourne
Lines: 28

>In article <[e-mail]>,
>[e-mail] writes:
>>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]
>>
>>
> WLOG assume the monic polynomial is irreducible over the rationals as
> this is the interesting case.
I meant the maximum of the absolute values of
  |\alpha_j - \alpha_i|  for all i,j.
Then one has C=\sqrt(3), etc. as in my previous posting. If one is interested
in the minimum of these absolute values then no such C exists but that is
another question.
Sorry about the confusion.

Bill Lloyd-Smith