Newsgroups: sci.math
From: hbaker@netcom.com (Henry Baker)
Subject: Re: Exposition of Budan's theorem ?
Date: Sun, 14 Dec 1997 16:27:49 GMT

In article <rbloomEL4BwH.49B@netcom.com>, rbloom@netcom.com (Ron Bloom) wrote:

> Can anyone find a concise exposition of Budan's theorem?
> In an old algebra text, dating from the 1930s, I find a 
> two and a half page argument, by the middle of which I 
> am about to doze off.
> 
> By the way, we were not taught any of these results in 
> any class, high school or college.  The method of 
> "Sturm Sequences" for determining the exact number of
> real roots of a polynomial, similarly, seems to have
> been neglected.

Since Budan's theorem doesn't handle all cases, Sturm's theorem
is more powerful.  On the other hand, Budan's theorem doesn't
require as much computation as Sturm's theorem does.

Routh-Hurwitz is the more-or-less culmination of Sturm's ideas to
finding roots in the complex plane.  It is used to characterize the
stability of linear systems.

Don't pooh pooh those older algebra texts.  Uspensky, Chrystal, Hamming,
Weisner, Lovitt, etc., cover a lot of stuff no longer available in modern
texts. 
The modern texts are even more of a snooze, since most deal _only_ in
abstractions -- they never give any concrete examples.

Some excellent books:

Dejon, B., and Henrici, P., eds.  Constructive Aspects of the Fundamental
Theorem of Algebra.  Wiley-Interscience, London, 1969.

Marden, M.  The Geometry of the Zeros of a Polynomial in a Complex
Variable.  Amer. Math. Soc., Providence, RI, 1949.

Two modern books you will find helpful:

Barbeau, E.J.  Polynomials.  Springer, 1989.  ISBN 0-387-96919-5.

Zippel, R.  Effective Polynomial Computation.  Kluwer, 1993. ISBN
0-7923-9375-9.

For Sturm sequences:

ftp://ftp.netcom.com/pub/hb/hbaker/quaternion/Sturm-FAQ.txt