I need to find analytically the roots of the following polynomial equation:
for an arbitrary integer $n$ and an arbitrary real parameter $a$. The only trick I can think of is in the special case when $a\geq0$ and $n=2m$ (i.e. $n$ even). In this case the equation can be decomposed into two simpler and lower dimensional equations:
but then I don't know how to proceed (I was thinking that maybe through some change of coordinates it is possible to transform these new equations into trinomial equations of the form $x^n-x+t=0$, whose solution is known, see here). I'd also like to find the roots for any $n$ and $a$. Closed-form solutions are not required: I'd really appreciate also a solution written e.g. as a series expansion.
Many thanks in advance.
P.S.: I don't know if it may help, but in my case $x\in[-1,1]$ so that you may write $x=\cos y$ for $y\in [0,\pi]$.