Ideal Projectors of Type Partial Derivative and Their Perturbations

TL;DR

This paper proves a conjecture about ideal projectors of a specific type, showing they can be approximated by perturbed Lagrange projectors, and provides an algorithm to compute the perturbation bound.

Contribution

It verifies Carl de Boor's conjecture for ideal projectors of partial derivative type and introduces an algorithm to determine the perturbation magnitude.

Findings

Existence of a positive perturbation bound for ideal projectors of partial derivative type.

Ideal projectors can be approximated by Lagrange projectors within this bound.

An algorithm to compute the perturbation magnitude based on Gröbner escalier.

Abstract

In this paper, we verify Carl de Boor's conjecture on ideal projectors for real ideal projectors of type partial derivative by proving that there exists a positive $\eta\in \mathbb{R}$ such that a real ideal projector of type partial derivative $P$ is the pointwise limit of a sequence of Lagrange projectors which are perturbed from $P$ up to $\eta$ in magnitude. Furthermore, we present an algorithm for computing the value of such $\eta$ when the range of the Lagrange projectors is spanned by the Gr\"{o}bner \'{e}scalier of their kernels w.r.t. lexicographic order.