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

Shuzo Izumi¹

¹ Research Center for Quantum Computing Kindai University Higashi-Osaka 577-8502, Japan

Annales Polonici Mathematici, Tome 113 (2015) no. 1, pp. 1-42.

Résumé

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 spaces of de Boor and Ron, there exists a projection $ {\mathsf T}_{\boldsymbol b}$ from the local ring $\mathcal {O}_{n,\boldsymbol b}$ onto the space $Z_{\boldsymbol b}$ of germs of elements of $Z$ at $\boldsymbol b$. At a general point $\boldsymbol b\in U$ its kernel is an ideal and $ {\mathsf T}_{\boldsymbol b}$ induces the structure of an Artinian algebra on $Z_{\boldsymbol b}$. In particular, this holds at points where the $k$th jets of elements of $Z$ form a vector bundle for each $k\in \mathbb {N}$. For an embedded manifold $X\subset \mathbb {C}^m$, we introduce a space of higher order tangents following Bos and Calvi. In the case of curve, using $ {\mathsf T}_{\boldsymbol b}$, we define the Taylor projector of order $d$ at a general point $\boldsymbol a\in X$, generalising results of Bos and Calvi. It is a retraction of $\mathcal {O}_{X,\boldsymbol a}$ onto the set of polynomial functions on $X_{\boldsymbol a}$ of degree up to $d$. Using the ideal property stated above, we show that the transcendency index, defined by the author, of the embedding of a manifold $X\subset \mathbb {C}^m$ is not very high at a general point of $X$.

Détail

DOI : 10.4064/ap113-1-1

Keywords: mathcal denote algebra holomorphic functions subset subset mathbb subset mathcal its finite dimensional vector subspace theory least spaces nbsp boor ron there exists projection mathsf boldsymbol local ring mathcal boldsymbol space boldsymbol germs elements boldsymbol general point boldsymbol its kernel ideal mathsf boldsymbol induces structure artinian algebra boldsymbol particular holds points where kth jets elements form vector bundle each mathbb embedded manifold subset mathbb introduce space higher order tangents following bos calvi curve using mathsf boldsymbol define taylor projector order general point boldsymbol generalising results bos calvi retraction mathcal boldsymbol set polynomial functions boldsymbol degree using ideal property stated above transcendency index defined author embedding manifold subset mathbb high general point nbsp