Can a finite polynomial have no roots on the left of the complex plane?

In summary, the conversation discusses the conditions for a finite polynomial P(x) to have no roots on the left of the complex plane defined by Re(x<0). The speaker suggests using a conformal map on the left of the complex plane and counting the zeroes using the integral of f'/f.
  • #1
zetafunction
391
0
given a finite polynomial

[tex] a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+...+a_{n}x^{n} =P(x)[/tex]

is there a theorem or similar to ensure that P(x) has NO roots on the left of complex plane defined by [tex] Re(x<0) [/tex] ??
 
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  • #2
What restrictions do you put on coefficients?

If you put n = 1, a_0 = 1, a_1 = 1, you get a polynomial with a root x=-1 on the left of complex plane right away. Perhaps I did not understand your question?
 
  • #3
I think he's asking what conditions make it so there are no roots with real part negative
 
  • #4
Ah! That makes sense.
 
  • #5
Map the left of the complex plane to the unit disk via a conformal map (you need the möbius map) and then count the zeroes by evaluating the integral of f'/f.
 

1. What are zeros on the complex plane?

Zeros on the complex plane refer to points on the plane where a function or equation equals zero. These points are also known as roots or solutions.

2. How are zeros on the complex plane different from real zeros?

Zeros on the complex plane can be any point on the plane, including imaginary numbers, while real zeros are restricted to the real number line.

3. How are zeros on the complex plane represented?

Zeros on the complex plane are typically represented as points on the plane, with the real component being the x-coordinate and the imaginary component being the y-coordinate.

4. What is the significance of zeros on the complex plane?

Zeros on the complex plane can help determine the behavior and solutions of a function or equation. They can also provide insight into the structure and properties of the complex numbers.

5. How are zeros on the complex plane calculated?

Zeros on the complex plane can be calculated using various methods, such as the quadratic formula or graphical analysis. It ultimately depends on the specific function or equation being analyzed.

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