Abelianization of Matrix Orthogonal Polynomials

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 Padé 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 Padé approximation on a Riemann surface and corresponding notion of biorthogonality of sections of the semicanonical bundle (half-differentials). Several examples are offered in illustration of the new notions.

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