Examine the polynomial 
$$
x^{\tau+1}-x^{\tau}+\alpha=0\,
$$
and denote by $\left|x_{\max}\left(\tau,\alpha\right)\right|$ as the
maximal magnitude of a root of this equation. For $\tau>1$, I observed numerically
that this root lies on the unit circle, i.e. 
$$
\left|x_{\max}\left(\tau,\alpha\right)\right|=1
$$
if and only if 
$$
\frac{1}{\alpha}= a\tau+b
$$
for some constants $a$ and $b$. This numerical relation ("prediction"), and it's linear fit could be seen here

[![enter image description here][1]][1]

I wonder if this linear relationship could be proved (at least in some limit). 

Thanks in advance!


  [1]: https://i.sstatic.net/O8keO.png