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

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.