It is addressed in the post that the equation $1+z^p+z^q=z^n$ have no multiple complex roots where $p<q<n$ (On the irreducibility of certain trinomials and quadrinomials).

Q. Let us consider the polynomial $\mathbf{P}(z)=1+z^p+z^q+z^r-z^n$ where $p,q,r$ and $n$ are natural numbers with $1<p<q<r<n$. Has \mathbf{P} any multiple complex root?