On primary decomposition of Hermite projectors

TL;DR

This paper investigates the structure of ideal projectors in polynomial spaces, providing a primary decomposition approach and characterizing when they are limits of Lagrange projectors, with an application to symmetric ideals.

Contribution

It introduces a primary decomposition framework for ideal projectors and characterizes their limits of Lagrange projectors, including a counterexample involving symmetric ideals.

Findings

Ideal projectors can be decomposed into sums with primary kernels.

A projector is a limit of Lagrange projectors if and only if each component is.

Constructed an example of a symmetric ideal projector not approximable by Lagrange projectors.

Abstract

An ideal projector on the space of polynomials $\mathbb{C} [\mathbf{x}]=\mathbb{C} [x_{1},\ldots ,x_{d}]$ is a projector whose kernel is an ideal in $\mathbb{C}[ \mathbf{x}]$. The question of characterization of ideal projectors that are limits of Lagrange projector was posed by Carl de Boor. In this paper we make a contribution to this problem. Every ideal projector $P$ can be written as a sum of ideal projector $\sum P^{(k)}$ $\ $such that $\cap \ker P^{(k)}$ is a primary decomposition of the ideal $\ker P$. We show that $P$ is a limit of Lagrange projectors if and only if each $P^{(k)}$ is. As an application we construct an ideal projector $P$ whose kernel is a symmetric ideal, yet $P$ is not a limit of Lagrange projectors.