We start with a naive definition.
A curve over a field k is the set of zeros in
k × k of a single polynomial equation in
two variables, f(x, y) = 0.
Let k=R and consider the following curves.
(Here we should insert the graphs of the following curves.)
These examples raise several important
questions about the
naive definition. For example
y = 0.
y2 = 0.
xy = 1.
x2 + y2 = 1.
x2 + y2 = 0.
x2 + y2 = -1.
y2 = x3.
y2 = x3 + x2.
y2 = x3 - x.
x2 - y2 = 0.
(y - x)(y - x + 1) = 0.
x3 + y3 = 1.
The questions are of two kinds. The first two questions are
metaphysical; they ask if we have the correct definition. The final
two questions are analytic; they begin to study the intrinsic geometry
- Are (i) and (ii) the same or different curves?
- Should (v) and (vi) even be called curves at all?
- Why are (iii), (ix) and (xi) disconnected?
- What's special about (vii), (viii), and (x)?
To elaborate on question (A), consider the families of curves
(Pictures again need to be inserted.)
Intersect each family of curves with the line x =
1. When a ≠ 0, curves in the first
family intersect the line in exactly one point, but curves in the
second family intersect in two points. As the parameter a
|→ 0, the limiting curve of the first family is
curve (i), and it still intersects in one point. However, the limiting
curve in the second family is curve (ii), and the two points of
intersection coalesce in the limit. Should the limit curve intersect
the line in one point, or in two points "properly counted"?
- y = ax.
- y2 = ax.
Regarding question (B), there are two schools of thought (which will
be described shortly). In either case, the following theorem shows
that the underlying difficulty is that R is not
Theorem Let k be an algebraically closed field and
let f ∈ k[x,y] be a nonconstant
polynomial. Then there are infinitely many solutions to
The restrictive school of thought says that you can avoid the
difficulties posed by examples (v) and (vi) by considering only
algebraically closed fields. This seems to be rather narrow-minded,
since it foregoes any chance of applying geometric techniques to some
interesting problems that arise in number theory. A more open-minded
approach is to "fix" the naive definition to make it useful over
arbitrary fields. Nevertheless, we will acquiesce (at least
temporarily), and work over algebraically closed fields for the
remainder of this chapter.
Write f(x,y) = a0(y) + a1(y)x + … +
, with an(y) ≠ 0
, as a
polynomial of degree n
coefficients from k[y]
. If n = 0
then we simply have f(x,y) = a0(y)
, a nonconstant
polynomial in one variable. Since k
algebraically closed, this polynomial has a root
. Then all of the infinitely many pairs
ranges over the
(infinite) algebraically closed field k
solutions to f(x,y) = 0
So, suppose n > 0. Then an(y) =
0 has only finitely many solutions. There are infinitely
many points y0 where an(y0) ≠
0; at each one of them, we get a polynomial
f(x,y0) of degree n in the
single variable x. This polynomial has a solution
x0, yielding infinitely many points
(x0,y0) where f(x,y) = 0.
As a set, affine n
-space over a field
is defined to be the Cartesian product
Ank = k × k × … × k
with itself n
Let k be algebraically closed. An
affine algebraic set over k is
the set of common zeros in
Ank of some set of polynomials S
⊂ k[x1, …, xn]. Given such a set
S, its algebraic set of zeros will be denoted
The Zariski topology on An is
defined by taking the closed sets to be the algebraic sets. The
Zariski toplogy on an algebraic set is defined as the topology induced
from its embedding in An.
Comments on this web site should be addressed to the
Kevin R. Coombes
Department of Biomedical Informatics
The Ohio State University
Columbus, Ohio 43210