Speaker
Description
Let $f$ be an algebraic function of degree $m+1$ and $f_\infty$ be its holomorphic germ at the point $\infty$. Hermite--Padé polynomials of type I for the tuple $[1,f_\infty,f_\infty^2, \dots,f_\infty^m]$ of order $n$ at $\infty$ are $m+1$ polynomials $Q_{n,j}$, $j=0,\dots,m$, such that $\deg Q_{n,j}\le n$ and
$$
Q_{n,0}(z)+Q_{n,1}(z)f_\infty(z)+Q_{n,2}(z)f_\infty^2(z)+\dots+Q_{n,m}(z)f_\infty^m(z)=O(z^{-m(n+1)})
$$
as $z\to\infty$.
In 1984 J. Nuttall (not in general case and not with full proofs) and in 2017 E. Chirka, R. Palvelev, S. Suetin and A. Komlov (in general case and with full proofs) showed that $Q_{n,m-1}/Q_{n,m}$ asymptotically recover the sum of the values of $f$ on first $m$ sheets of Nuttall partition of the Riemann surface of $f$. So, this ratio recovers sum of $m$ values of $(m+1)$-valued function $f$.
In 2021 the polynomial Hermite--Padé $m$-system was introduced. With the help of this system we show that for generic function $f$ the ratio of some minors of size $m+1-k$ of the $(m+1)\times(m+1)$ matrix consisting of Hermite--Padé polynomials of order $n, n-1,\dots, n-m$ asymptotically recover the sum of the values of $f$ on first $k$ sheets of Nuttall partition of the Riemann surface of $f$ for each $k=1,\dots,m$. Hence we constructivelly recover $m$ values of $(m+1)$-valued algebraic function $f$.
