From: israel@math.ubc.ca (Robert Israel)
Subject: Re: Convexity of set of polynomial roots
Date: 12 Jan 2001 00:01:33 GMT
Newsgroups: sci.math
Summary: Polynomials with only large roots do not form a convex set

In article <93le4l$10d$1@nnrp1.deja.com>, skyblur  <skyblur@my-deja.com> wrote:

>Let P(z) = 1 + c1*z + ... + cn*z^n
>Define P(z) to be "stable" if all roots have |z| > 1.
>Let C(n) be the *closure* of the set of c1...cn for which
>P(z) is stable.  For example C(1) is the interval [-1, 1]
>and C(2) is the closed triangle with vertices (2,1), (-2, 1)
>and (0, -1).  Is there some theorem which guarantees C(n) to
>be convex (in c1 ... cn) for all n?

No, it isn't true for n=3.  Consider for example the polynomials

P1(z) = 1 + (2 a/3) z + (a^2/3) z^2 - a^3 z^3
P2(z) = 1 + (4 a/3) z + a^2 z^2 - (a^3/3) z^3

where a = 0.851.  Then according to Maple, P1(z) and P2(z) are stable (the 
minimum absolute values of roots being approximately 1.0069 and 1.0002 
respectively), but (P1(z) + P2(z))/2 is not, the minimum absolute
value being approximately 0.9996.

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2