Let 
and 
be univariate polynomials in a complex variable
,
and let the polynomial degrees of 
and 
satisfy 
. Then
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
is a simple closed clockwise-oriented contour, 
is the set of roots of 
inside of 
, and 
is the set of roots of 
outside of 
.
The first equality is an instance of the residue theorem. On the Riemann sphere, the simple closed contour 
splits the sphere into two regions. After the change of variables 
, the point zero is mapped to infinity and vice versa.
What was the "inside" of 
becomes the outside of 
in the new coordinate. The second equality is the residue
theorem applied to the meromorphic one-form 
in the coordinate 
, with a minus sign because 
travels clockwise after the coordinate change. The hypothesis
on the degrees of 
and 
ensure that 
does not have a pole at 
.
The above diagram shows two different points of view of the contour 
and the poles of the meromorphic
one-form 
on the Riemann sphere. The usual point of view
is centered at 
,
but the role of inside and outside is switched from the point of view of 
. The poles inside are labeled blue and outside are green.
The theorem also follows from taking the contour integral at infinity, i.e., a circle of large radius 
. The hypothesis on the degree says that this integral tends
to zero. Hence it must actually be zero, because at some point the circle contains
all of the poles of 
. This is a special case of the fact that on a compact Riemann surface, in this case the Riemann
sphere, the sum of the complex residues of
a meromorphic one-form is zero.
