Inside-Outside Theorem

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 poleat .

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.