Algebraic Aspects of Polynomial Interpolation in Several Variables

This paper summarizes relations between the constructive theory of polynomial ideals and polynomial interpolation in several variables. The main ingredient is a generalization of the algorithm for division with remainder to \quotients" of polynomials and nite sets of polynomi-als. x1. Introduction When compared to the univariate case, polynomial interpolation in several variables turns out to be notoriously troublesome by providing various non-trivial new diiculties. To overcome some of these problems and to gain understanding of polynomial interpolation in several variables, it is necessary to adopt and apply techniques from algebraic geometry, in particular of the constructive theory of polynomial ideals. The main ingredient of this paper will be a concept closely related to the so{called Grr obner bases, which have been introduced by Buchberger 5,6] in 1965 and which are an important tool in all Computer Algebra systems, especially, but not only, for algebraic techniques to solve polynomial systems of equations. The issue to be considered here is a very simple observation in the uni-variate case: let x 0 ; : : : ; x n 2 IR be distinct points and let !(x) = (x ? x 0) (x ? x n) be the polynomial of degree n + 1 with normalized leading coeecient which vanishes in all the points. It is well{known and probably taught in any class on Numerical Analysis that for any f : fx 0 ; : : : ; x n g ! IR there exists a unique interpolation polynomial L x 0 ;:::;x n f of degree n. Moreover, if p is any polynomial, then there is an \algebraic" way to compute the interpolant with Approximation Theory IX ISBN 1-xxxxx-xxx-x. All rights of reproduction in any form reserved.