Analytic continuation of algebraic functions by using Hermite–Padé polynomials of first kind

Let $f$ be an algebraic function and $f_0$ some its germ at a point $x_0$. A natural question is how and in what domain can one reconstruct the values of the initial function $f$ by its germ $f_0$? All rational approximants, the Padé approximants for example, reconstruct the initial function $f$ in those domains where the germ $f_0$ can be continued as a singlevalued holomorphic function. But our function $f$ is multivalued. How does one reconstruct the other values of $f$?
In the talk we will consider the reconstruction of the values of $f$ by using Shafer quadratic approximants. These approximants are built by the Hermite–Padé polynomials of first kind, which naturally generalize the Padé polynomials. We will see that in the case of a three-valued function $f$, Shafer quadratic approximants reconstruct at once two values of $f$ at every point of the plane.