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

[zetafunction]

given a finite polynomial

$$a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+...+a_{n}x^{n} =P(x)$$

is there a theorem or similar to ensure that P(x) has NO roots on the left of complex plane defined by $$Re(x<0)$$ ??

 

[hamster143]

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?

 

[Office_Shredder]

I think he's asking what conditions make it so there are no roots with real part negative

 

[hamster143]

Ah! That makes sense.

 

[Count Iblis]

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.