Polynomial Hermite-Pad\'e $m$-system for meromorphic functions on a compact Riemann surface

TL;DR

This paper introduces a polynomial Hermite-Padé m-system for meromorphic functions on compact Riemann surfaces, analyzing its asymptotic zero distribution and polynomial ratios, with applications to functions with specific algebraic structures.

Contribution

It develops a new Hermite-Padé m-system framework for meromorphic functions on Riemann surfaces and studies its asymptotic behavior and zero distribution.

Findings

Zeros of polynomials have a specific limit distribution.

Ratios of polynomials converge to sums of function values on Riemann surface sheets.

Results apply to functions with algebraic relations, like powers of a meromorphic function.

Abstract

For an arbitrary tuple of $m+1$ germs of analytic functions at a fixed point, we introduce the so-called polynomial Hermite-Pad\'e $m$-system (of order $n$, $n\in\mathbb N$), which consists of $m$ tuples of polynomials; these tuples, which are indexed by a natural number $k\in[1,\dots,m]$, are called the $k$th polynomials of the Hermite-Pad\'e $m$-system. We study the weak asymptotics of the polynomials of the Hermite-Pad\'e $m$-system constructed at the point $\infty$ from the tuple of germs $[1, f_{1,\infty},\dotsc$, $f_{m,\infty}]$ of the functions $1, f_1,\dots,f_m$ that are meromorphic on some $(m+1)$-sheeted branched covering $\pi\colon \mathfrak R\to\widehat{\mathbb C}$ of the Riemann sphere $\widehat{\mathbb C}$ of a compact Riemann surface $\mathfrak R$. In particular, under some additional condition on $\pi$, we find the limit distribution of the zeros and the asymptotics of the ratios of the $k$th polynomials for all $k\in[1,\dots, m]$. It turns out that in the case, where $f_j = f^j$ for some meromorphic function $f$ on $\mathfrak R$, the ratios of some $k$th polynomials of such Hermite-Pad\'e $m$-system converge to the sum of the values of the function $f$ on the first $k$ sheets of the Nuttall partition of the Riemann surface $\mathfrak R$ into sheets.