What is the Number of Complex Roots for the Given Equation with Varying d?

In summary, the conversation discusses the complex roots of a fourth degree equation and how the number of roots varies with the value of d. The equation is simplified to a maximum of 8 zeroes, but it is unclear which of these are actual roots of the original equation. The speaker has found that for small values of d, only four roots are possible, while for larger values, two more complex conjugate roots appear. The question is asked if there are any analytical tools to determine the critical value of d where these additional roots appear.
  • #1
eradi
5
0
How many complex roots admit the following equation:
(2 z^2 + 1)^2 ((z + d)/(z - i))^1/2 - (2 z^2 - d)^2 ((z + i)/(z - d))^1/2 == 0
for 0 < d < 1, where i = (-1)^1/2.
Can I found how their number varies with d by using the argument principle?
Thanks in advance for helpfull suggestions
 
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  • #2
You get a fourth deg. equation in z^2 equation on simplification, so a maximum of 8 zeroes is possible.
 
  • #3
Yes, but which of them are actually zeroes of the starting equation?
Thanks for your interest
 
  • #4
I found numerically that for small d only four roots are admissible (two reals and two purely imaginary) but for large d two more complex conjugate roots appear.
There exists any analitical tools to define this critical value of d?
 

1. What is the definition of "number of complex roots"?

The number of complex roots refers to the number of solutions or values of a polynomial equation that contain the imaginary number i. Complex roots appear in pairs, with each pair consisting of a complex number and its complex conjugate.

2. How can the number of complex roots be determined?

The number of complex roots can be determined by looking at the degree of the polynomial equation. The degree of the equation is equal to the number of complex roots. For example, a quadratic equation with a degree of 2 will have 2 complex roots.

3. What does it mean if a polynomial equation has no complex roots?

If a polynomial equation has no complex roots, it means that all of its solutions are real numbers. This can be seen when the discriminant of the equation (b²-4ac) is greater than or equal to 0, indicating that there are no imaginary solutions.

4. Can a polynomial equation have more complex roots than its degree?

No, a polynomial equation can only have a maximum number of complex roots equal to its degree. This is because the fundamental theorem of algebra states that a polynomial equation of degree n will have exactly n complex roots, counting multiplicities.

5. How do complex roots affect the graph of a polynomial function?

Complex roots can affect the graph of a polynomial function by creating a point of inflection or a local minimum/maximum at the point where the root is located. This can be seen as a change in the concavity or direction of the curve at that point.

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