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

 

[blue_raver22]

I would like to clarify that the term "zeros" in this context refers to the values of z that make the expression $1+z^{2^n}$ equal to zero. This is also known as finding the roots of the equation.

Upon examining the expression, it is clear that $1+z^{2^n}$ does not have any zeros on the unit disc. This is because for any value of z on the unit disc, the term $z^{2^n}$ will always have a magnitude less than or equal to 1. When we add 1 to this term, the resulting magnitude will always be greater than 1. Since the unit disc has a radius of 1, it is not possible for the expression to equal zero on this domain.

Furthermore, as n increases, the term $z^{2^n}$ will become increasingly small, making it even more impossible for the expression to have zeros on the unit disc. This can be seen by substituting different values of n into the expression and observing the resulting graph on the unit disc.

In conclusion, the reason why $1+z^{2^n}$ does not have zeros on the unit disc is due to the nature of the expression itself and the properties of the unit disc. This is a mathematical phenomenon that can be explained and understood through rigorous analysis and experimentation.