From: Stefan Loewe
Subject: Re: arbitrary-precision real arithmetic
Date: 
Message-ID: <STEFAN.95Aug30095643@euclid.math.nat.tu-bs.de>
In article <··········@fsuj01.rz.uni-jena.de> ···@rz.uni-jena.de (Ralf Muschall) writes:
   In article <·······················@192.0.2.1> ······@netcom.com (Henry Baker) writes:
   :: I forgot to say that for algebraic numbers (roughly solutions of polynomials
   :: with integer coefficients -- e.g., sqrt(2)), there is a very straightforward
   :: way to calculate with them -- by representing such an algebraic number by a
   :: square matrix whose characteristic polynomial is the minimal polynomial which
   :: has the given algebraic number as a root.

   :: The Cayley-Hamilton theorem tells you that one of these matrices satisfies
   :: its own characteristic polynomial, so you can now calculate with the set
   :: of square matrices generated by one of these particular matrices.

   A layman's question:

   Wouldn't this approach map all zeros of the polynomial to the same
   [set of] matri[x|ces]?

   I.e., sqrt(2) and (-sqrt(2)) would be indistinguishable.
   This might not be a problem in pure algebra, where they
   *are* indistinguishable, but in other sciences, the users might
   want to know whether a number is +1.414 or -1.414.

   Ralf

1) If we represent sqrt(2) as [[0,2],[1,0]], we have -sqrt(2) = -[[0,2],[1,0]],
so the two roots of X^2 - 2 ARE different,

BUT

2) to compare the elements of our field {[[a,2b],[b,a]] : a, b rational} by
size, we need to embed it into the real numbers, and there are TWO embeddings,
one mapping [[0,2],[1,0]] to 1.414..., the other mapping it to -1.414...

To make things unique one has to decide about the representation of sqrt(2) as
a matrix ( [[0,2],[1,0]] or -[[0,2],[1,0]]), and about the embedding.

Stefan
From: Bob Silverman
Subject: Re: arbitrary-precision real arithmetic
Date: 
Message-ID: <424lo5$6sf@puff.mathworks.com>
In article <····················@euclid.math.nat.tu-bs.de>,
Stefan Loewe <······@euclid.math.nat.tu-bs.de> wrote:
:In article <··········@fsuj01.rz.uni-jena.de> ···@rz.uni-jena.de (Ralf Muschall) writes:
:   In article <·······················@192.0.2.1> ······@netcom.com (Henry Baker) writes:
:   :: I forgot to say that for algebraic numbers (roughly solutions of polynomials
 
stuff deleted....

:   A layman's question:
:
:   Wouldn't this approach map all zeros of the polynomial to the same
:   [set of] matri[x|ces]?
:
:   I.e., sqrt(2) and (-sqrt(2)) would be indistinguishable.
 
Not really. For any finite extension of the rationals (say Q(a)) where a
is an algebraic number whose minimal polynomial is f,  a and conj(a) are
distinguished by the Galois group, i.e. the action of the Galois group
Gal(a/Q) sends a to -a.  

To distinguish a from -a in practice, (say on a computer), you simply
must specify which embedding to choose, i.e. how to embed a in C.  This
can be done by simply giving the argument of a in the complex plane,
along with its magnitude.

-- 
Bob Silverman
The MathWorks Inc.
24 Prime Park Way
Natick, MA