Repeated Roots of Polynomials Whose Coefficients are Either 0 or 1 Consider a polynomial of the form: $$f\left(z\right)=1+z^{p_{0}}+z^{p_{1}}+\ldots+z^{p_{N}}$$ where the $p_{n}$s are distinct positive integers. Are the roots of $f$ (in $\mathbb{C}$) necessarily simple (i.e., must they all have multiplicities of $1$)?

No.  We can construct as follows:  The polynomial $(1+z^a)(1+z^b)$ will have two factors of $1+z$ if $a$ and $b$ are both odd.  So if we choose $b$ a good bit bigger than $a$, the cross terms will miss each other and all coefficients will be $1$. For instance,
$$f(z) = (1+z^3)(1+z^5) = 1+z^3+z^5+z^8$$
has $-1$ as a double root.
