Third degree polynomial

The applet demonstrates the effect of the polynomial mapping f(Z) = (Z-A)(Z-B)(Z-C). The applet displays two copies of the complex plane, the left one representing the domain of f and the right one its codomain. The domain contains all three zeros of f, A, B and C. You can drag these points around to change the polynomial.

The domain also contains a circle Γ, with center D, passing through a point E. You can change the radius of the circle by draging E, or its center by draging D. You can also drag the actual circle around. There is a red point F which you can drag around the circle.

The right plane contains the image of Γ under the mapping f, and also the point G = f(F). If you change or move the circle or move the points A, B or C, you will see that the curve f(Γ) changes accordingly. If you drag the point F around the circle Γ, you will see the point G tracing the curve f(Γ).

Finally, the left plane also contains a "free" point H. You can move this point anywhere on the plane without constrains. Its image in the right plane is I.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and activated. (click here to install Java now)

Suggestions:

  • Finding p-points: click on the button to set up the applet for this activity. You can also set it up by hand to obtain another similar configuration. Find the winding number of f(Γ) around the point I=f(H). How many other points will be mapped to I? Can you find them by dragging the point H around untill the I returns back to its current position?
  • Cube roots: click on the button to set up the applet. This will set A = B = C = 0, and make Γ into the unit circle. That means f(Z) = Z3 and f(Γ) is also the unit circle, but wrapped around three times. By dragging the F around Γ, can you find all three cube roots of 1? How about cube roots of i?
  • Cusps: click on the button to set up the applet. This will move B to A so now A will be a double zero of f. It will also make Γ pass through A. Look at the shape of f(Γ) at the origin. You should see a pointy shape that is usually called a cusp. Can you explain why the cusp is there? Why does an image of a perfectly smooth curve suddenly have a cusp? Drag F around Γ to figure out which point on the curve does the cusp come from. Is it a zero of f? What is the multiplicity of that zero? Does that help you explain what's going on?
  • This is a continuation of the previous activity. When you think you understand what's going on, slowly start dragging E so that the circle Γ slowly grows. Watch carefuly f(Γ). You should see the casp at the origin disappear, and, little bit later, another cusp appear at different place. When that happens, stop dragging E, and instead drag F around the circle until you know which point on Γ does this new cusp comes from. Is there something interesting about the location of this point? Slowly drag H in a circle around this point, and observe f(H). Can you explain what's going on here? What is the significance of the cusps? Look at and attempt exercise 4 on page 370 in the textbook.
  • Multiplicities: click on the button to reset the applet to its original state. Now all the zeros of f have multiplicity 1 again. Does f have any p-points pf multiplicity 2? How about 3? How many of them are there? Can you find them by dragging the circle and the points F and H around? If there are no p-points of multiplicity 3, can you move the points A, B and C so that f will still have three simple zeros, but at least one p-point of multiplicity 3?

Jan Hlavacek, Created with GeoGebra