It is addressed in the  [post][1] 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][2]). 

>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? 


  [1]: https://mathoverflow.net/questions/447134/can-the-equation-1zz2-zn-have-multiple-complex-roots
  [2]: https://doi.org/10.7146/math.scand.a-10593