Groebner basis structure of ideal interpolation

TL;DR

This paper explores the connection between Groebner bases and ideal interpolation, introducing a new concept and an efficient algorithm for computing Groebner bases of polynomial ideals related to interpolation problems.

Contribution

It introduces the notion of 'reverse' complete reduced basis and presents a fast algorithm for computing Groebner bases of ideal projectors' kernels under various orderings.

Findings

The proposed algorithm efficiently computes Groebner bases for ideal projectors.

Knowing the affine variety aids in determining the reduced Groebner basis.

The study clarifies the relationship between Groebner bases and ideal interpolation conditions.

Abstract

We study the relationship between certain Groebner bases for zero dimensional ideals, and the interpolation condition functionals of ideal interpolation. Ideal interpolation is defined by a linear idempotent projector whose kernel is a polynomial ideal. In this paper, we propose the notion of "reverse" complete reduced basis. Based on the notion, we present a fast algorithm to compute the reduced Groebner basis for the kernel of ideal projector under an arbitrary compatible ordering. As an application, we show that knowing the affine variety makes available information concerning the reduced Groebner basis.