Just a brief remark that if $M=2$ and the constant term is $\pm2$, then these are called *Garsia numbers*. It is known that $z=1$ is a limit point for this set (and some computational results as well). Perhaps, you'll find the [following recent paper][1] useful as far as the techniques are concerned. 


  [1]: http://www.math.uwaterloo.ca/~kghare/Preprints/PDF/P36_Garsia.pdf