Sketch of a Proof by Birkhoff and MacLane
Consider the polynomial P(z) as a mapping from one copy of the complex plane (say, the z-plane) to another copy of the complex plane (say, the w-plane).
Such a mapping transforms a circle |z| = r of radius r
into a closed curve on the w-plane (see Figure 1). For very large values of r,
the a_{n}z^{ n} term dominates, and the image is a closed curve
which loops n times around the origin of the w-plane (see Figure 2). On the other
hand, for very small values of r, we may neglect all the terms except
a_{m}z^{ m} + a_{0},
where a_{m} is the coefficient with the lowest index
m>0 which is not equal to zero, and the image loops m times around the point
w = a_{0} (Figure 3). As
r0, the image collapses to the point
a_{0}. We now invoke the topological fact that if we
start with r very large, and continuously reduce r to 0, simultaneously reducing
the image in the w -plane from a large loop encircling the origin (likely more than
once) to a point, we must necessarily encounter a stage where the image curve passes through the
origin w = 0. Thus, we have shown that some point on the z-plane maps to
Figure 1. Image of circle |z| = 2 under the mapping P(z) = z^{3} - 2z^{2} + z - 1. | Figure 2. Image of circle |z| = 20 under the mapping P(z) = z^{3} - 2z^{2} + z - 1. |
Figure 3. |z| = 0.1 under the mapping P(z) = z^{3} - 2z^{2} + z - 1. Note the enlarged scale. | Figure 4. Image of circle |z| = 0.75 under the mapping P(z) = z^{3} - 2z^{2} + z - 1. Apparently a root of this polynomial lies very near this circle. |
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