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Computing Groebner bases of ideal interpolation

Xue Jiang, Yihe Gong

TL;DR

The paper addresses computing the reduced Gröbner basis of the vanishing ideal for finite sets of points under ideal interpolation. It introduces a formal power series framework, leveraging Taylor expansions to translate interpolation functionals into series and then reads off the reduced Gröbner basis by Gaussian elimination, yielding a polynomial-time algorithm. The approach unifies interpolation with Gröbner-basis computation, offers a general strategy that encompasses multi-point and multiplicity cases, and connects to the MMM algorithm via a single-point reduction when advantageous. The results provide concrete procedures and examples, including a special decomposition that can improve efficiency in practice.

Abstract

We present algorithms for computing the reduced Gröbner 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 Gröbner 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.

Computing Groebner bases of ideal interpolation

TL;DR

The paper addresses computing the reduced Gröbner basis of the vanishing ideal for finite sets of points under ideal interpolation. It introduces a formal power series framework, leveraging Taylor expansions to translate interpolation functionals into series and then reads off the reduced Gröbner basis by Gaussian elimination, yielding a polynomial-time algorithm. The approach unifies interpolation with Gröbner-basis computation, offers a general strategy that encompasses multi-point and multiplicity cases, and connects to the MMM algorithm via a single-point reduction when advantageous. The results provide concrete procedures and examples, including a special decomposition that can improve efficiency in practice.

Abstract

We present algorithms for computing the reduced Gröbner 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 Gröbner 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.
Paper Structure (6 sections, 4 theorems, 46 equations, 2 figures, 3 algorithms)

This paper contains 6 sections, 4 theorems, 46 equations, 2 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Computing a "reverse" reduced team by Gaussian elimination
  4. Interpolation monomial basis and quotient ring basis
  5. The algorithms to compute the reduced Gröbner bases
  6. A special example for ideal interpolation

Key Result

Lemma 5

Given interpolation conditions $\Delta = \delta_{\bf 0} \circ {\rm span}\{P_1(D), P_2(D), \dots, P_n(D)\}$, where $P_1, P_2, \dots, P_n \in \mathbb{F}[{\bf X}]$ are linearly independent polynomials. Let $T=\{{\bf X}^{\boldsymbol{\beta_1}}, {\bf X}^{\boldsymbol{\beta_2}}, \dots, {\bf X}^{\boldsymbol{

Figures (2)

  • Figure 1: Lagrange interpolation
  • Figure 2: Ideal interpolation

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4: $\prec$-minimal monomial basis SauerT1998
  • Lemma 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Example 9
  • Example 10