Computing the monodromy for a cover of the Riemann sphere (and Puiseux expansions)

The potential ramification points are $X_0$ s.t. either $P$ and $\partial P/\partial Y$ have a common zero at some $(X_0,Y_0)$, or the degree of $P(X_0,Y)\in\mathbb{C}[Y]$ is less than $n$, or $X_0=\infty$. Let us consider the first case, i.e. suppose that $Y_0\in\mathbb{C}$ is a $k$-fold root of $P(X_0,Y)\in\mathbb{C}[Y]$ (the remaining cases are similar). For $X_1$ close to $X_0$ this multiple root will split into $k$ different roots. Generically we will get a $k$-cycle in the local monodromy at $X_0$ (the roots undergo a cyclic permutation when we go around $X_0$). Sometimes we may, however, get a different permutation of these $k$ roots. It can be determined using Newton polygon as follows.

We may suppose that $(X_0,Y_0)=(0,0)$ (by shifting the variables). For every monomial $Y^aX^b$ in $P$ we draw the point $(a,b)$ in the plane and then we take the convex hull of these points. We consider only the part of the boundary of the resulting polygon from which $(0,0)$ is visible. Let $s_1,\dots,s_q$ be these sides, and let $k_1,\dots,k_q$ be the lengths of the projections of $s_i$'s to the $x$-axis; we clearly have $k_1+\dots+k_q=k$. If none of the sides $s_i$'s contain any integer points inside then $(k_1,\dots,k_q)$ are the lengths of the cycles of the local monodromy.

For example, for the polynomial $X^3+ 5XY^3-8Y^7+iY^{10}+(2+i)X^4-X^2Y^4$ we get $-8Y^7+iY^{10}$ when we set $X=0$, i.e. $Y=0$ is a $7$-fold root, and the Newton polygon is

so that we get a $3$-cycle and a $4$-cycle.

(If some of $s_i$ contain an integer point inside then things become more complicated: essentially we know that we have a Puiseux series solution $Y=cX^t+\dots$ of $P(X,Y)=0$, where $-t$ is the slope of $s_i$, and we need to determine what denominators appear in the exponents of this series. We basically pose $Z=Y-cX^t$, get an equation for $Z$, and use again Newton polygon to see what happens.)

Newton polygon is equally useful for determining ramifications for finite extensions of $p$-adic numbers (the method is the same).