Hermite–Padé polynomials of type I for a tuple of meromorphic functions on a compact Riemann surface and Shafer approximants

For a natural number $m$, let $\mathcal R$ be a compact Riemann surface with the fixed $(m+1)$-sheeted branched covering $\pi:\mathcal R\to\hat{\mathbb C}$ of the Riemann sphere $\hat{\mathbb C}$. Let $f_1,f_2,\dots,f_m$ be meromorphic functions on $\mathcal R$ such that the functions $1$, $f_1$, …, $f_m$ are linearly independent over $\mathbb C(z)$. We consider the Hermite–Padé polynomials of type I for the tuple of germs $[1;f_{1,\infty};f_{2,\infty};\dots;f_{m,\infty}]$, where $f_{j,\infty}$, $j=1,\dots,m$, are generated by the germs of $f_j$ at the same point in $\pi^{-1}(\infty)$. We study the asymptotic behaviour of these polynomials and discuss how to use them for the (asymptotic) reconstruction of values of an algebraic function $f$ of degree $m+1$ from its given germ (for such reconstruction we should choose $f_j:=f^j$). In particular, for the case $m=2$ we consider so-called quadratic Shafer approximants for a germ of an algebraic function of degree 3. The talk is based on the joint work with E.M.Chirka, A.V.Komlov, and S.P.Suetin.