28 - Polynomial interpolation of Minimal Degree and Gröbner Bases

Summary

Abstract

This paper investigates polynomial interpolation with respect to a finite set of appropriate linear functionals and the close relations to the Gröbner basis of the associated finite dimensional ideal.

Introduction

In the 33 years since their introduction by Buchberger (1965, 1970), Gröbner bases have been applied successfully in various fields of Mathematics and to many types of problems. This paper wants to go the opposite way by presenting a different approach to Gröbner bases for zero dimensional ideals from the quite recent theory of polynomial interpolation of minimal degree. The latter one is an approach introduced by de Boor and Ron (1990, 1992) to solve interpolation problems defined by a finite number of linear functionals using appropriate polynomial spaces with certain useful properties.

Let me briefly explain this with the example of Lagrange interpolation in ℝ^d. Suppose that a finite set of pairwise disjoint points {x₀, …, x_N} ∈ ℝ^d is given. The Lagrange interpolation problem consists of finding, for any y₀, …, y_N, a polynomial p such that p(x_j) = y_i, j = 0,…, N. Clearly, this problem is always solvable and even has infinitely many solutions. The “real” question, however, is to find a polynomial subspace P such that for any given data the Lagrange interpolation problem has a unique solution in P and to choose P “as simple as possible”.