[SOLVED]zeros on D(0,1)

dwsmith

Well-known member
Why doesn't $1+z^{2^n}$ have zeros on the unit disc?

CaptainBlack

Well-known member
Why doesn't $1+z^{2^n}$ have zeros on the unit disc?
All its zeros are on the unit circle, aren't they?

CB

dwsmith

Well-known member
All its zeros are on the unit circle, aren't they?

CB
I don't think so. If we solve for z, we have $z = (-1)^{1/2^n}$

CaptainBlack

Well-known member
I don't think so. If we solve for z, we have $z = (-1)^{1/2^n}$
$$z^{2n}=-1=e^{(2k+1)\pi i}, k \in \mathbb{Z}$$

so:

$$z=e^{\frac{(2k+1)\pi}{2n}\;i}, k \in \mathbb{Z}$$

of which $$2n$$ are distinct, but all lie on the unit circle.

CB