I am given a power series $T(z) = t_nz^n +$ some higher order term (of degrees not being multiples of $n$), $n>1$ and all $t_i \in \mathbb{N}$. I am trying to solve for $z$ in the unit disc (contained in the convergence disk) an equation $T(z) = \epsilon$, for some $\epsilon$ very close to zero. If $n$ was 1, this would be series reversion, but I am not sure how to extend this result here (if possible).
What I tried so far: if $T$ was $t_nz^n$, then an immediate solution would be $z^n = \epsilon/t_n$. This tells me that I should look for a series in the form $S(\epsilon) = \epsilon/t_n +s_2\epsilon^2 + \dots$ such that $S(T(z)) = z^n$ (or something similar, up to constants maybe). I guess one could work the coefficients, but this does not feel like a proof. In particular, I do not know how to guarantee the convergence of such an $S$. As I only need the first order term of $S$, my first question would be: are there known results to say that such an $S$ with rational coefficients exists, or converges somewhere in the unit disk, provided it exists?
If $S$ exists, then $T(S(z)) = t_n^{1-n}z^n + \dotsc,$ so that with $f(Y) = T(Y)-t_n^{1-n}z^n$, I have $f(S(z)) \equiv 0 \bmod z^{n+1}$. I then tried to apply Hensel's lemma to $f(Y) \in \mathbb{R}[[z]][[Y]]$, but of course here the derivative vanishes. Now I am out of ideas --- this is not really my usual field. Because of the context where I am trying to obtain this result, I am quite convinced that there is a way to make a rigourous proof with formal argument, but I am not sure if I am using the correct tools. My second question is: Are there variants of Hensel's lemma that could help me here, or other kind of techniques? Google and keywords do not seem to help (yet).
My last question is: does someone know sources on similar problems, or could give hints on how to proceed?