Thomas Sauer. Abstract. This paper summarizes ..... Twain, M., The Adventures of Tom Sawyer, Hartford 1876. Thomas Sauer. Mathematical Institute. University ...
Algebraic aspects of polynomial interpolation in several variables Thomas Sauer Abstract. This paper summarizes connections between the constructive theory of polynomial ideals in several variables and multivariate polynomial interpolation. The main ingredient is a reasonable generalization of the division–with–remainder algorithm to multivariate polynomial ideals.
§1. Introduction Compared to the univariate case, polynomial interpolation in several variables is known to be notoriously troublesome by providing a lot of nontrivial new difficulties. To overcome some of these problems and to gain some 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 Gr¨ 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 univariate case: let x0 , . . . , xn ∈ IR be distinct points and let ω(x) = (x − x0 ) · · · (x − xn ) be the polynomial of degree n + 1 which vanishes in all the points. It is well–known and probably teached in any class on Numerical Analysis that for any f : {x0 , . . . , xn } → IR there is always a unique interpolation polynomial Lx0 ,...,xn f of degree n. Moreover, if p is any polynomial, then there is a second, “algebraic” way to compute the interpolant with respect to p by doing division with remainder. Indeed, the polynomial p can be uniquely decomposed into p(x) = q(x)ω(x) + r(x), Approximation Theory IX Charles K. Chui and Larry L. Schumaker (eds.), pp. 1–3. Copyright c 1998 by Vanderbilt University Press, Nashville, TN. ISBN 1-xxxxx-xxx-x. All rights of reproduction in any form reserved.
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where r is a polynomial of smaller degree than ω, i.e.,a polynomial of degree n. Since q(x)ω(x) vanishes at x ∈ {x0 , . . . , xn }, the remainder polynomial r coincides with p there and hence r = Lx0 ,...,xn p. There are some simple remarks on this “algebraic approach” to interpolation: first, this approach immediately carries over to multiple interpolation points and yields a Hermit interpolation polynomial then. Second, it is possible to interpolate at points which are given only implicitly as the zeros of a polynomial; the points themselves do not have to be computed. In several variables this observation becomes much more involved. First, a finite set of distinct points does not define a “generic” associated interpolation space and the structure of this interpolation space depends not only on the number of points, but also on their geometric configuration (cf. [2,4,10]). Second, the multivariate decomposition corresponding to (1) needs additional assumptions on the “quotient”. And, finally, not even the notion of “degree” is generic or unique in the multivariate setting. We fix some notation for multivariate polynomials. Let IK be an infinite field and let Π = IK [x1 , . . . , xd ] denote the ring of polynomials in d variables over IK. This ring is equipped with a graded structure by means of a totally ordered monoid Γ, i.e.,an (additive) semigroup with neutral element (zero) and an admissible well–ordering “
