Spaces of polynomial functions of bounded degrees on an embedded manifold and their duals

TL;DR

This paper develops a framework for polynomial functions on embedded manifolds, introducing Taylor projectors and higher order tangents, and analyzes their algebraic properties and embeddings.

Contribution

It generalizes the Taylor projector concept to embedded curves and manifolds, and establishes new results on the algebraic structure of polynomial functions on these spaces.

Findings

Defined Taylor projectors of order d on embedded curves.

Introduced higher order tangents for embedded manifolds.

Proved a zero-estimate indicating limited transcendental complexity.

Abstract

Let $\mathcal{O}(U)$ denote the algebra of holomorphic functions on an open subset $U\subset\mathbb{C}^n$ and $Z\subset\mathcal{O}(U)$ its finite-dimensional vector subspace. By the theory of least space of de Boor and Ron, there exists a projection $T_b$ from the local ring $\mathcal{O}_{n,b}$ onto the space $Z_b$ of germs of elements of $Z$ at $b$. At general $b\in U$, its kernel is an ideal and induces a structure of an Artinian algebra on $Z_b$. In particular, it holds at points where $k$-th jets of elements of $Z$ form a vector bundle for each $k\le\dim_{\mathbb{C}}Z_b-1$. Using $T_b$ we define the Taylor projector of order $d$ on an embedded curve $X\subset\mathbb{C}^m$ at a general point $\boldsymbol{a}\in X$, generalising results of Bos and Calvi. It is a retraction of $\mathcal{O}_{X,a}$ onto the set of the polynomial functions on $X_a$ of degree up to $d$. For an embedded manifold $X\subset\mathbb{C}^m$, we introduce a set of higher order tangents following Bos and Calvi and show a zero-estimate for a system of generators of the maximal ideal of $\mathbb{C}\{t-b\}$ at general $b\in X$. It means that $X$ is embedded in $\mathbb{C}^n$ in not very highly transcendental manner at a general point.