Perturbing the Constant Coefficient of a Complex Polynomial
Perturbing the Constant Coefficient of a Complex Polynomial
This Demonstration plots the complex polynomial . For , the multiplicity of a root determines the number of cycles (cyan, blue, yellow, red) around the root (shown as a black dot). An example is shown in the first snapshot. You can vary , , , , , and . Small changes in the constant term, which makes , transforms the polynomial to a form (z-), with all of the roots of the new polynomial being simple.
P(z)=+a
m
1
(z-)
z
1
m
2
(z-)
z
2
m
3
(z-)
z
3
a=0
z
1
z
2
z
3
m
1
m
2
m
3
a
a≠0
∏
k
z
k