We are interested in studying the algebraic geometry around singular points of $w(z)$ given implicitly by
$$
\begin{equation}
f(z,w)=a_0(z)+a_1(z)w+a_2(z)w^2+\cdots+a_n(z)w^n
\end{equation}
$$
with $z,w\in \mathbb{C}$ and $a_i(z)$ polynomials with rational coefficients. $w(z)$ is $n$-valued except at singular points. In this discussion, we use the following definition of a singular point: We know for a function $w(z)$ given implicitly by $(1)$, we have
$$w'(z)=-\frac{f_z}{f_w}$$
and so we find the points $(z,w)$ where $f(z,w)=f_w(z,w)=0$. We can find these points by first determining where the resultant $R(f,f_w)=0$. The resultant is a polynomial in $z$ the zeros of which are the $z$-values of the singular points. Finding $z$, we then solve for $w$ by plugging into $f(z,w)=0$. This is easily done in Mathematica with the code:

`thezvals = z /. NSolve[Resultant[theFunction, D[theFunction, w], w] == 0, z]`

`thewvals = NSolve[{theFunction == 0, D[theFunction, w] == 0} /. z -> #] & /@ thezvals`

Once we have the singular points, we then identify the branching geometry around these points by computing power expansions of the branches.

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