Hermite–Padé polynomials in the model case

By the model case we mean the situation where we consider Hermite–Padé polynomials for a tuple $[1, f_0, f_0^2]$, where $f_0$ is a germ of 3-valued algebraic function $f$. This model case was first time investigated by J. Nuttall in 1981. In 1984 he showed how to recover two values of our 3-valued algebraic function $f$ with the help of Hermite–Padé polynomials in question. But his proofs had gaps and were given not in general case. These disadvantages were removed in the joint work by E. Chirka, R. Palvelev, S. Suetin, A. Komlov in 2017. In the talk we discuss this result and other results on the properties of these Hermite–Padé polynomials. In particular, we discuss asymptotic properties of the corresponding discriminants (joint result with R. Palvelev) and the interpolation points.