A coordinate-free proof of the finiteness principle for the Whitney extension problem

TL;DR

This paper introduces a coordinate-free proof of the finiteness principle for Whitney's extension problem, utilizing ideals and translation-invariant subspaces, emphasizing the role of compactness in the proof.

Contribution

It provides a novel, coordinate-free approach to Whitney's extension problem, simplifying the proof by using algebraic and geometric concepts instead of induction.

Findings

Coordinate-free proof of the finiteness principle

Use of ideals and translation-invariant subspaces

Highlighting the importance of compactness in the proof

Abstract

We present a coordinate-free version of Fefferman's solution of Whitney's extension problem in the space $C^{m-1,1}(\mathbb{R}^n)$. While the original argument relies on an elaborate induction on collections of partial derivatives, our proof uses the language of ideals and translation-invariant subspaces in the ring of polynomials. We emphasize the role of compactness in the proof, first in the familiar sense of topological compactness, but also in the sense of finiteness theorems arising in logic and semialgebraic geometry.