Zero distribution for Angelesco Hermite--Pad\'e polynomials

E. A. Rakhmanov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 73 (2018) no. 3, pp. 457-518

Résumé

This paper considers the zero distribution of Hermite–Padé polynomials of the first kind associated with a vector function $$ \vec f=(f_1,\dots,f_s) $$ whose components $f_k$ are functions with a finite number of branch points in the plane. The branch sets of component functions are assumed to be sufficiently well separated (which constitutes the Angelesco case). Under this condition, a theorem on the limit zero distribution for such polynomials is proved. The limit measures are defined in terms of a known vector equilibrium problem. The proof of the theorem is based on methods developed by Stahl [59]–[63] and Gonchar and the author [27], [55]. These methods are generalized further in the paper in application to collections of polynomials defined by systems of complex orthogonality relations. Together with the characterization of the limit zero distributions of Hermite–Padé polynomials in terms of a vector equilibrium problem, the paper considers an alternative characterization using a Riemann surface $\mathcal R(\vec f\,)$ associated with $\vec f$. In these terms, a more general conjecture (without the Angelesco condition) on the zero distribution of Hermite–Padé polynomials is presented. Bibliography: 72 titles.

Détail

Keywords: rational approximations, Hermite–Padé polynomials, zero distribution, equilibrium problem, $S$-compact set.