From: wcw@math.psu.edu (William C Waterhouse)
Subject: Re: algebraic topology and real closed fields
Date: 17 May 2000 21:30:21 GMT
Newsgroups: sci.math
Summary: [missing]

In article <8fnm67$10q67@hkunae.hku.hk>, 
Siu Lok Shun <lssiu@hkusua.hku.hk> writes:
> Allan Adler <ara@nestle.ai.mit.edu> wrote:
> > Siu Lok Shun <lssiu@hkusua.hku.hk> writes:
> ... 
> > > The results (i) and (ii) are used by Wood (1985) to prove the existence of 
> > > singular eigenvalue of quaternion matrices. If my guess is true, then his 
> > > result can be generalized to any quaternion algebra over real closed 
> > > field. But is it true? Would some algebraic topologist give me an
> > > affirmative answer? TIA.
> 
> > For the application you have mind, however, you probably don't need to
> > transfer the homotopy theoretic results. You can probably formulate
> > Wood's result directly as an elementary statement (for each n) and
> > transfer it to other real closed fields.
> 
> Sorry, I am afraid I can't follow. Wood made a strong result of algebraic
> topology to assure the existence of a number q_0 such that A-q_0 I is
> singular. Such number q_0 is not constructed from any elementary mean.


This has not been clarified yet, so I'll try to explain the issue.

Take an n. Consider the statement 

   For every n by n matrix A with quaternion entries, there is some
   further quaternion  q  and some n-tuple of quaternions v such
   that v is not identically zero and Av = vq.  

[The question for Av = qv is different but would be treated in the same way.] 
 
If we write this out by expressing each of the quaternions in A, v, q
as a sum of real multiples of 1,i,j,k, then we find that this statement
is an assertion of the following form: 

    We have a number of parameters  t_1, t_2, ...  (Here, they are
    coefficients of the entries in A.)  We have some additional 
    variables  x_1, x_2, ... (Here, they are the coefficients in q and 
    v.)  We have a certain collection of (integral) polynomials involving 
    the  t's and the x's. (Here, they are the coefficients in Av-vq and
    the sum of squares of the coefficients in v.)  Then for all values of 
    the t's, there are values of the x's such that certain of these 
    polynomials (the coefficients in Av-vq) are zero and others 
    (here, just the sum of the squares of the coefficients in v)   
    are nonzero.

This is the sort of "elementary" statement covered by "Tarski's 
Principle", which says that it is true in all real closed fields 
if it is true in any one real closed field.  Hence if you can prove
the theorem for the reals (no matter what methods you use), then you
know it is automatically true for all other real closed fields.

I think most modern presentations of Tarski's principle are bound up with
related (though not terribly difficult) logic concepts, but there is
a fairly complete treatment from scratch in Chapter 5 of N. Jacobson, 
Basic Algebra I. 

William C. Waterhouse
Penn State