Let $X$ be a smooth, projective and geometrically connected curve of genus $g\geq 3$ defined over $\mathbb C$. Suppose the function field $k(X)$ of $X$ takes the form $k(X)=\mathbb C(x)[y]$, where $$y^a=f(x)$$ with $a\geq 3$ an integer and with $f\in \mathbb C[x]$ a separable polynomial of degree $d=\deg(f)\geq 3$. 


Question: a) Is it possible to give an upper bound for $d$ and $a$ in terms of $g$?


b) Is there even a precise formula relating $a,d,g$?



Relation of degree and genus of superelliptic curves
$2g=da+ma\_1-2a-d-m+2$, where $a=a\_1m,$ and $m$ is the g.c.d ($d,a$).


Edit.
Let $\chi(S)=2-2g$ be the Euler characteristic. Hurwitz formula gives
$$\chi(S)=2a-r,$$
where $r$ is the ramification: a branch point of order $k$ contributes
$k-1$ to $r$. As your polynomial has $d$ simple roots, these roots
contribute $(a-1)d$ to the ramification. To find ramification at infinity
we write our equation $w^a=P\_d(z)$ as $w^a=z^dh(z)=u^d$,
where $h$ is a holomorphic function at $\infty$,
$h(\infty)\neq 0$, and $u$ a germ of a 
meromorphic function with a simple pole at $\infty$.
The last equation factors into irreducible factors:
$$w^a-u^d=\prod\_c (w^{a\_1}-cu^{d\_1}),$$
where $c$ are the roots of unity of degree $m$, $m$ is the greatest
common factor of $a$ and $d$, $d=md\_1,$ and $a=ma\_1$.


From this we conclude that our Riemann surface $S$ has $m$ ramification
points of order $a\_1$ at $\infty$. So we obtain
$$2-2g=2a-(a-1)d-m(a\_1-1).$$


