Subresultants, Sylvester sums and the rational interpolation problem

TL;DR

This paper introduces a novel approach to univariate rational interpolation using subresultants, providing explicit formulas for Cauchy interpolation and determinantal expressions for osculatory interpolation, advancing computational methods in this area.

Contribution

It offers new explicit formulas and determinantal expressions for rational interpolation problems using subresultants, extending previous results and providing practical computational tools.

Findings

Explicit formulas for Cauchy interpolation in terms of symmetric functions.

Determinantal expressions for osculatory rational interpolation.

Connections to previous matrix formulations by Beckermann and Labahn.

Abstract

We present a solution for the classical univariate rational interpolation problem by means of (univariate) subresultants. In the case of Cauchy interpolation (interpolation without multiplicities), we give explicit formulas for the solution in terms of symmetric functions of the input data, generalizing the well-known formulas for Lagrange interpolation. In the case of the osculatory rational interpolation (interpolation with multiplicities), we give determinantal expressions in terms of the input data, making explicit some matrix formulations that can independently be derived from previous results by Beckermann and Labahn.