The fundamental theorem of algebra, by winding numbers

Gauss’s 1799 dissertation gave the first substantial proof that every polynomial of degree \(n\) has \(n\) complex roots. The modern topological proof counts windings: on a huge circle \(p(z) \approx z^n\) winds \(n\) times around 0, on a tiny circle about a non-root it winds 0 times, so as the circle grows the image curve must pass through 0 – and the argument principle says it does so exactly \(n\) times.

import matplotlib.pyplot as plt
import numpy as np

from mathematicskit.complex_analysis import argument_principle, circle_contour

coefficients = [1.0, -2.0, 3.0, 1.0, -4.0, 2.0]  # degree 5
p = np.poly1d(coefficients)
print("roots:", np.round(np.roots(coefficients), 4))

roots: [ 0.8363+1.6276j  0.8363-1.6276j -1.0563+0.j      0.6918+0.2946j
  0.6918-0.2946j]

Winding number of \(p(|z| = R)\) about 0 as R grows

radii = [0.2, 0.6, 1.0, 1.4, 2.0, 5.0]
fig, axes = plt.subplots(2, 3, figsize=(11, 7))
for ax, R in zip(axes.ravel(), radii, strict=False):
    contour = circle_contour(0.0, R)
    image = p(contour.points(2000))
    winding = argument_principle(p, contour)
    inside = int(np.sum(np.abs(np.roots(coefficients)) < R))
    print(f"R = {R:3.1f}: winding number {winding}, roots inside {inside}")
    ax.plot(image.real, image.imag, lw=1)
    ax.plot(0, 0, "r+", ms=12, mew=2)
    ax.set_title(f"R = {R}: winds {winding} times")
    ax.set_aspect("equal")
fig.suptitle("Image of |z| = R under a degree-5 polynomial")
fig.tight_layout()

plt.show()

R = 0.2: winding number 0, roots inside 0
R = 0.6: winding number 0, roots inside 0
R = 1.0: winding number 2, roots inside 2
R = 1.4: winding number 3, roots inside 3
R = 2.0: winding number 5, roots inside 5
R = 5.0: winding number 5, roots inside 5