Constructive reconstruction of values of an algebraic function via polynomial Hermite–Padé $m$-system

Let $f_0$ be a given germ of some algebraic function $f$ of degree $m+1$. We consider the following problem: how to reconstruct constructively the values of f in “as large a region as possible” on its Riemann surface $\mathfrak R$?
We introduce the polynomial Hermite–Padé $m$-system, which, in our case, is constructed from the tuple of the germs $[1, f_0,f_0^2,\dots,f_0^m]$. This system includes the Hermite–Padé polynomials of the first and of the second type.
We show how to reconstruct the values of $f$ on the first $m$ sheets of the so-called Nuttall partition of the Riemann surface $\mathfrak R$ via the polynomial Hermite–Padé $m$-system.