From: chambert@math.jussieu.fr (Antoine Chambert-Loir)
Subject: Re: Zero-sets of polynomials
Date: 22 Jun 2000 09:00:03 -0500
Newsgroups: sci.math.research
Summary: [missing]

In article <vkaaegebsat.fsf@gs2.sp.cs.cmu.edu>,
 <from_usenet@andrej.com> wrote:
>Suppose P(x_1, ..., x_n) is a multivariate polynomials with rational
>coefficients. Let Z be the zero-set of P, i.e., a subspace of the
>Euclidean space R^n,
>Let A be the field of algebraic numbers. Question: is Z intersected
>with A dense in Z? In other words, is there an algebraic solution of
>P = 0 arbitrarily close to any real solution of P = 0?

Yes.
You can prove it along the following lines :

Noether's normalization lemma shows that there exists
polynomials  P_1,..., P_d (with d=n-1 in your case) with
rational coefficients such that the induced morphism 
    f : Z -> R^{n-1}
is finite. From the commutative algebra point of view,
the extension of Q-algebras
      Q[ P_1,...,P_d ] \subset k[X_1,...,X_n]/(P)
is entire (even finite).

This imply that 
  1/ if f(z) has algebraic coordinates, z already has.
  2/ if  f(z') is close from f(z), some point above f(z') is 
close from z  (properness of a finite morphism)
  3/ f is surjective

Now, take any z in Z. f(z) is an element of R^n which can be
appxoximated by some point x' with algebraic coordinates.
By 2/ and 3/, there exists a point z' above x' which is close
from  z.


-- 
Antoine Chambert-Loir  o-o  Institut de Mathématiques de Jussieu (Chevaleret)
Tel: 01 44 27 75 20     |   Université Pierre et Marie Curie
Fax: 01 44 27 48 44     -   Boite 247, 4 place Jussieu, F-75252 Paris Cedex 05
http://www.math.jussieu.fr/~chambert
==============================================================================

From: wcw@math.psu.edu (William C Waterhouse)
Subject: Re: Zero-sets of polynomials
Date: 22 Jun 2000 16:30:10 -0500
Newsgroups: sci.math.research

In article <vkaaegebsat.fsf@gs2.sp.cs.cmu.edu>, 
from_usenet@andrej.com writes:
>...
> In particular, I am interested in the case when Z is the set of
> singular n x n real matrices. In that case, is it true that already
> the set of rational singular matrices is dense in the set of all
> singular matrices?

Yes, this case has a simple proof.  In a singular matrix, one
column will be a linear combination of the others; without loss
of generality, let's suppose C_n = r_1 C_1 + ... + r_{n-1} C_{n-1}.
Choose  s_i to be good rational approximations to the r_i and
D_i (for i < n) to be good rational approximations to the C_i.
Set D_n = s_1 D_1 + ... + s_{n-1} D_{n-1}.


William C. Waterhouse
Penn State