Recovery of branch points of an algebraic function via Hermite–Padé polynomials

Let $f$ be an algebraic function and $f_\infty$ be some its germ at the point $\infty$. The Hermite–Padé polynomials of type I for the tuple $[1,f_\infty,f_\infty^2]$ of order $n\in\mathbb N$ are nontrivial polynomials $Q_{n,j}$, $j=0,1,2$, such that $\deg Q_{n,j}\leq n$ and the following relation is satisfied:
$$ Q_{n,0}(z)+Q_{n,1}(z)f_\infty(z)+Q_{n,2}(z)f_\infty^2(z)=O(z^{-2n-2}),\quad\text{ as }z\to\infty. $$
In 2015 some molecular chemists stated without correct justification that the zeros of the discriminants $D_n(z)$ of the quadratic equation
$$ Q_{n,0}(z)+Q_{n,1}(z)w+Q_{n,2}(z)w^2=0 $$
asymptotically reconstruct (as $n\to\infty$) branch points of $f$. We clarify and justify this statement in the model case when $f$ is an algebraic function of degree 3.
This is a joint work with Roman Palvelev.