Error Formulas for Ideal Interpolation

TL;DR

This paper explores the algebraic structure of error formulas in ideal interpolation, introducing 'normal' error formulas and demonstrating their existence via Gröbner bases, extending prior concepts and providing explicit examples.

Contribution

It introduces 'normal' error formulas for ideal interpolation and proves their existence for all cases using Gröbner bases, generalizing previous 'good' error formulas.

Findings

Existence of 'normal' error formulas for all ideal interpolations

Extension of de Boor's 'good' error formula to a broader class

Explicit form of 'normal' error formula for Shekhtman's example

Abstract

In this paper we study the algebraic structure of error formulas for ideal interpolation. We introduce the so-called "normal" error formulas and prove that the lexicographic order reduced Gr\"obner basis admits such a formula for all ideal interpolation. This formula is a generalization of the "good" error formula proposed by Carl de Boor. Finally, we discuss a Shekhtman's example and give an explicit form of "normal" error formula for this example.