Minimal Solutions of the Rational Interpolation Problem

We explore connections between the approach of solving the rational interpolation problem via resolutions of ideals and syzygies, and the standard method provided by the Extended Euclidean Algorithm (EEA). As a consequence, we obtain explicit descriptions for solutions of minimal degrees in terms of the degrees of elements appearing in the EEA. This result 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.