Complex Polynomial Mappings

Show . Set $k = -1$ $k = 1$


A complex number $x + iy$ may be identified with an ordered pair $(x, y)$ of real numbers. A complex polynomial consequently defines a mapping from the Cartesian plane to itself. The grids above represent the domain (left) and target (right) planes of a complex polynomial. The heavy grid consists of unit squares. Moving the mouse over the domain creates a disk in the domain. The image of this disk is plotted in the target.

The polynomials $f(z) = \frac{1}{n} z^{n} + kz$ have critical points where $f'(z) = z^{n-1} + k = 0$. These $(n - 1)$ points are uniformly placed around a circle of radius $\sqrt[n-1]{k}$ centered at the origin. When the domain disk encloses a critical point, the image of the boundary acquires a loop. A tangent vector traversing the image of the boundary turns once plus the number of critical points enclosed.