The polynomial Hermite-Padé $m$-system for meromorphic functions on a compact Riemann surface

Given a tuple of $m+1$ germs of arbitrary analytic functions at a fixed point, we introduce the polynomial Hermite-Padé $m$-system, which includes the Hermite-Padé polynomials of types I and II. In the generic case we find the weak asymptotics of the polynomials of the Hermite-Padé $m$-system constructed from the tuple of germs of functions $1, f_1,\dots,f_m$ that are meromorphic on an $(m+1)$-sheeted compact Riemann surface $\mathfrak R$. We show that if $f_j = f^j$ for some meromorphic function $f$ on $\mathfrak R$, then with the help of the ratios of polynomials of the Hermite-Padé $m$-system we recover the values of $f$ on all sheets of the Nuttall partition of $\mathfrak R$, apart from the last sheet.
Bibliography: 18 titles.