Abelianization of Matrix Orthogonal Polynomials

TL;DR

This paper explores the connection between matrix biorthogonal polynomials and scalar multi-point Padé approximation on Riemann surfaces, introducing new notions and illustrating them with examples.

Contribution

It introduces a novel framework linking matrix polynomial biorthogonality with scalar Padé approximation on Riemann surfaces, expanding the theoretical understanding.

Findings

New notion of scalar multi-point Padé approximation on Riemann surfaces

Definition of biorthogonality of sections of semi-canonical bundles

Several illustrative examples provided

Abstract

The main goal of the paper is to connect matrix polynomial biorthogonality on a contour in the plane with a suitable notion of scalar, multi-point Pad\'e approximation on an arbitrary Riemann surface endowed with a rational map to the Riemann sphere. To this end we introduce an appropriate notion of (scalar) multi-point Pad\'e\ approximation on a Riemann surface and corresponding notion of biorthogonality of sections of the semi-canonical bundle (half-differentials). Several examples are offered in illustration of the new notions.