MHB Finding the Zeros of $1+z^{2^n}$ on the Unit Disc

[Dustinsfl]

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

 

[CaptainBlack]

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

 

[Dustinsfl]

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]

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