## How to determine the genus and the number of poles in a ring $\mathbb{C}[t,t^{-1},u]/\langle u^m-p(t)\rangle$?

A pole of the finite polynomial $$p(t)=\sum_{i\in\mathbb{Z}}a_it^i$$ is a point $$p_0$$ such that $$p(p_0)=\infty$$.

I already know that when $$m=2$$, the genus $$g$$ depends on the defining polynomial $$p(t)$$ with degree $$d$$ according to the formula $$g=\frac{d-1}{2},$$ equivalently $$2g=\begin{cases} d-2 \textrm{ if d is even} \d-1 \textrm{ if d is odd.}\end{cases}$$

After that, I have that the number $$n$$ where poles are allowed depends on $$p(t)$$ according to the formula $$n=4-r$$ where $$r$$ is the number of ramified points in $$\{0,\infty \}$$: $$0$$ is ramified exactly when the constant term $$a_0=0$$, and $$\infty$$ is ramified exactly when the degree $$d$$ is odd. Combining this information gives $$2g+n-1=\begin{cases} d+1 \textrm{ if } a_0\neq0, \d-1 \textrm{ if } a_0=0. \end{cases}$$

As I said, it is the case when $$m=2$$. When $$m$$ is arbitrary, what could I say?

How to determine the genus and the number of poles in a ring $$\mathbb{C}[t,t^{-1},u]/\langle u^m-p(t)\rangle$$?