Minimal solutions of the rational interpolation problem

TL;DR

This paper links ideal resolution and syzygy methods with the Extended Euclidean Algorithm to explicitly describe minimal degree solutions in rational interpolation and polynomial parametrizations.

Contribution

It provides explicit formulas for minimal degree solutions using the EEA, connecting algebraic and algorithmic approaches.

Findings

Explicit descriptions of minimal solutions in terms of EEA degrees

Characterization of minimal degree in μ-bases for polynomial parametrizations

Bridging algebraic ideal theory with computational algorithms

Abstract

We explore connections between the approach of solving the rational interpolation problem via resolutions of ideals and syzygies with the standard method provided by the Extended Euclidean Algorithm. As a consequence, we obtain explicit descriptions for solutions of "minimal" degrees in terms of the degrees of elements appearing in the EEA. This allows us to describe the minimal degree in a $\mu$-basis of a polynomial planar parametrization in terms of a "critical" degree arising in the EEA.