Abstract
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 μ-basis of a polynomial planar parametrization in terms of a critical degree arising in the EEA.
Highlights
Let K be a field, l, n1, . . . , nl positive integers, n := n1 + · · · + nl, and (xi, yi,j) ∈ K2, for i = 1, . . . , l, j = 0, . . . , ni − 1, (1.1)with xi = xj if i = j
We are requiring that the rational function y(x) is defined on all the points xi, i = 1, . . . , l
Rational interpolation has been well studied in the last centuries, with references going back to the mid 1800’s ([5, 14, 12])
Summary
Rational interpolation; Syzygies; Extended Euclidean Algorithm; minimal degree; μ-basis. If one is interested in a fast method to solve any instance of the interpolation problem, the EEA is the most efficient tool available, and any improvement in dealing with this problem will get translated into a faster algorithm to solve the EEA In this sense, the two situations (solving the rational interpolation problem and computing the EEA of two polynomials) are equivalent from an algorithmic and complexity point of view. The two situations (solving the rational interpolation problem and computing the EEA of two polynomials) are equivalent from an algorithmic and complexity point of view We conclude this paper by applying our tools to study the κ-degree in Section 5, where we recover the results of Antoulas in [2] with our methods
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