If I have a function $f(z,\alpha)$ (let's keep it a polynomial of order $\geq 2$ in $z$, for simplicity), what would be a necessary condition for there to be branch points for this function? A friend mentioned that $f(z,\alpha)$ and $f\_{\alpha}(z,\alpha)$, where the second term is the partial derivative w.r.t. $\alpha$, should have a common root. I don't see this intuitively. Is this true? If so, why?



Necessary condition for a branch point
Your question is not very clear. However, I guess you are asking for the branch points of the cover of $\mathbb{C}$ defined by $f(z, \alpha)=0$.


In this case, let us assume for the sake of symplicity that $f(z, \alpha)$ is *monic* in $z$; then 


$f(z, \alpha)=z^n + f\_{n-1}(\alpha)z^{n-1}+ \cdots + f\_0(\alpha)$. 


The Riemann surface $X \subset \mathbb{C}^2$ defined by $f(z, \alpha)=0$ is a cover $X \to \mathbb{C}$ of degree $n$, defined by $(z, \alpha) \to \alpha$.


The branch points of this cover are precisely the points $\bar{\alpha}$ such that $f(z, \bar{\alpha})$ has a multiple root. This is equivalent to require that $f(z, \bar{\alpha})$ and $f\_z(z, \bar{\alpha})$ have a common root.


In other words, the branch points are the roots of the equation


$\textrm{Disc}\_z(f)=0,$


where $\textrm{Disc}\_z(f)$ is the discriminant of $f$ with respect to $z$, i.e. the resultant of $f$ and $\partial f / \partial z$ computed with respect to $z$.


For instance, the branch points of the Riemann surface defined by


$z^3+f\_1(\alpha)z+f\_0(\alpha)=0$


are the roots of


$4 f\_1(\alpha)^3+27 f\_0(\alpha)^2=0$.


