Computing Groebner bases of ideal interpolation

TL;DR

This paper introduces a polynomial-time algorithm for computing reduced Groebner bases of vanishing ideals in ideal interpolation, translating interpolation conditions into formal power series and outperforming existing methods in certain cases.

Contribution

The paper presents a novel polynomial-time algorithm that computes Groebner bases for ideal interpolation by translating conditions into formal power series and using Gaussian elimination.

Findings

Algorithm has polynomial time complexity.

Outperforms MMM algorithm in specific ideal interpolation cases.

Effective for both single and multiple point ideal interpolation.

Abstract

We present algorithms for computing the reduced Gr\"{o}bner basis of the vanishing ideal of a finite set of points in a frame of ideal interpolation. Ideal interpolation is defined by a linear projector whose kernel is a polynomial ideal. In this paper, we translate interpolation condition functionals into formal power series via Taylor expansion, then the reduced Gr\"{o}bner basis is read from formal power series by Gaussian elimination. Our algorithm has a polynomial time complexity. It compares favorably with MMM algorithm in single point ideal interpolation and some several points ideal interpolation.