The Set of Unattainable points for the Rational Hermite Interpolation Problem

TL;DR

This paper characterizes the set of points where the Rational Hermite Interpolation Problem has no solution, describing its algebraic and geometric structure and providing algorithms for its computation.

Contribution

It introduces a detailed geometric and algebraic description of unattainable points in the Rational Hermite Interpolation Problem, including algorithms and equations for their identification.

Findings

The unattainable points form a union of equidimensional complete intersection varieties.

The number of these varieties equals the minimum of the degrees of numerator and denominator.

Each variety decomposes into rational subvarieties corresponding to input data points.

Abstract

We describe geometrically and algebraically the set of unattainable points for the Rational Hermite Interpolation Problem (i.e. those points where the problem does not have a solution). We show that this set is a union of equidimensional complete intersection varieties of odd codimension, the number of them being equal to the minimum between the degrees of the numerator and denominator of the problem. Each of these equidimensional varieties can be further decomposed as a union of as many rational (irreducible) varieties as input data points. We exhibit algorithms and equations defining all these objects.