From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Algebra fans: describing the inverse image of a polynomial
Date: 8 Feb 1996 02:44:01 GMT

In article <4fb25m$j3a@pucaix.rdc.puc-rio.br> you write:

>considerer the polynomial
>
>$$
>p(x, y, z) = \sum_{i = 0}^{n} a_{i}(x, y) z^{i}.
>$$
>
>where $x$, $y$ and $z$ are real variables. I would like to describe
>the surface in $R^{3}$ given by the set
>
>$$
>p^{-1}(x, y, z) = \{(x, y, z) \in R^{3} \, | \, p(x, y, z) = 0\}.
>$$

This would normally be written  p^{-1} (0).  I'll call this set  X.

>Is this a Riemann surface? Topological properties?

What it is (sort of by defintion) is a real algebraic set, and a (hyper)surface
in  R^3. There are many things it could be:

empty (e.g.  p=x^2+y^2+z^2+1)
a simgle point (e.g.  p=x^2+y^2+z^2)
a curve  (e.g.  p = z^2+(x^2+y^2-1)^2)
a "surface" (This is the generic case. 
   Note that  e.g. p=xyz  gives "singular points") In particular, it could be:
a 2-dimensional manifold (this occurs if there is no (x,y,z) in  X  where
	all three partial derivatices vanish simultaneously.)
a union of sets of this type (e.g.  p = p1*p2*p3... .Note: if  p  is not a
	product, then  X  is called an algebraic variety.)

There are variations you can discuss. For example, X  could be bounded
(e.g. p=x^2+y^2+z^2-1) or not (e.g. z=x^2+y^2-z^2-1), it could have
"holes" (e.g. p=z^2+(x^2+y^2-1)(x^2+y^2-2) ), and so on.

I don't know that I would say there is consensus as to what consitutes a
Riemann surface. For me, it always involves a complex-analytic structure,
which this  X  does not have in a natural way.

dave