From: kramsay@aol.commangled (Keith Ramsay)
Subject: Re: Theory of equations in several variables
Date: 26 Aug 2000 04:25:33 GMT
Newsgroups: sci.math
Summary: [missing]

In article <8o1fo9$hkm$1@nnrp1.deja.com>, j_tyler@my-deja.com writes:
|> If you mean polynomial equations in several variables, then there
|> is a theory of equations and systems of equations. It's
|> called algebraic geometry (since these systems define curves, surfaces
|> etc. in n-dimensional space) and it is HUGE.
...
|  For me, math is a hobby, not an occupation, and I am much more
|familiar with the history of math than modern math.  I simply had in
|mind an extension of the classical theory of equations, as in Galois.

Parts of algebraic geometry are bear some relation to Galois theory,
and then of course there are direct applications.

In Galois theory, one considers how roots of polynomials permute under
the Galois group. One can ask the same kind of question about the set
of common roots of a set of polynomials, and there's quite a lot which
has been written various aspects of it.

Here's a question which Grothendieck once described. (I'm afraid I
don't remember whether he was the first.) Suppose we have a set of
polynomials in several variables with coefficients of the form
a+b*sqrt(N) for some fixed integer N, where a and b are rational, and
consider their common roots (z_1,...,z_m) where z_i are complex
numbers. Assume there are no singularities. (I don't want to bother
with a precise definition, but basically there's a tangent plane to
each point of the right dimension; no pointy bits.) :-) Does the set
of equations you get by conjugating all the coefficients (from
a+b*sqrt(N) to a-b*sqrt(N)) ever have a different *topology*? I hope
I haven't left out any important conditions here. It apparently is
a substantial problem; Mazur found it interesting once but didn't
solve it.

We can dispose of the case N<0 right away, for example, because
substituting a-b*sqrt(N) for a+b*sqrt(N) in that case is just taking
complex conjugates of the coefficients. Complex conjugation taking
(z_1,...,z_m) to (z_1*,...,z_m*) is a continuous mapping taking each
solution of the one set of equations to a solution to the other set
and vice-versa, so they have the same topology.

If it were the real solutions we were considering, we could
get an easy counterexample like x^2+(1+sqrt(2))*y^2=1 versus
x^2+(1-sqrt(2))*y^2=1, where the former is an ellipse and the
latter is a hyperbola. But the sets of complex number solutions of
these two equations are topologically equivalent.

Some parts of algebraic geometry are a bit closer to what was known
classically as the "theory of equations". Resultant and resolvant
theory has gotten some charming extensions in modern times, even
though it had a reputation for having been pretty vigorously "mined
out" a long time ago.

Keith Ramsay