From: James Buddenhagen <jbuddenh@texas.net>
Subject: Re: Roots of the derivative of a polynomial
Date: Thu, 11 Jan 2001 04:27:00 GMT
Newsgroups: sci.math
Summary: Roots of f' lie in the convex hull of roots of f

Walsh in AMS Colloquium Pub's vol XXXIV refers to this theorem as
"Lucas's theorem" and mentions that it follows from an 1816 theorem of
Gauss.  The Lucas reference is:  F. Lucas 1874 Paris Comptes Rendus, vol
78, pp 271-274.  There are several other similar theorems, one called
Walsh's theorem.
--Jim Buddenhagen

Lee Rudolph wrote:
> 
> Robin Chapman <rjchapman@my-deja.com> writes:
> 
> >In article <93hek5$etl$1@nnrp1.deja.com>,
> >  Will Self <wself@msubillings.edu> wrote:
> >> There is a famous theorem that the roots of the derivative of a
> >> polynomial with complex coefficients lie in the convex hull of the
> >> roots of the polynomial itself.  Can someone tell me what this theorem
> >> is called and give me a reference to it?  Thanks.
> 
> [proof omitted]
> 
> I could swear I learned this as "a theorem of Walsh", but the last
> time I taught complex analysis, using Bak and Newman's _Complex
> Analysis_, I saw that the authors attributed it to Gauss.  (At
> least, I think so.  At the moment I can't find it in their book.
> Bummer.)
> 
> Lee Rudolph