Reconstruction of the values of an algebraic function via the system of Hermite-Padé polynomials

In the talk we suggest a method of reconstruction of the values of an algebraic function from its initial germ on all sheets of its Riemann surface, except for the “last” (see below), with the help of systems of linear algebraic equations. More precisely, for a given germ $f_0$ of an algebraic function $f$ of order $(m + 1)$, for each natural number $n$ we define a system of $m$ tuples of polynomials. These tuples are numbered by the number $k = 1, ..., m$, and we call them "$k$-th polynomials of Hermite-Padé $m$-system (of order $n$)". All these polynomials are found constructively, as solutions of linear homogeneous systems, and coefficients of these systems are some linear combinations of the Taylor coefficients of the original germ $f_0$. It turns out that the ratio of some polynomials from the $k$-th set assymptotically (as $n\to\infty$) reconstructs the sum of the values of the original function $f$ on the first $k$ sheets of the so-called Nuttall partition of Riemann surface of $f$ into sheets. We note that $1$-th polynomials of Hermite-Padé $m$-system are well-known Hermite–Padé polynomials of the second type and $m$-th polynomials of Hermite-Padé $m$-system are well-known Hermite–Padé polynomials of the first type.