PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

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

Volume 113 / 2015

Shuzo Izumi Annales Polonici Mathematici 113 (2015), 1-42 MSC: Primary 32B15; Secondary 11J82, 13J07, 32E30, 41A10, 41A63. DOI: 10.4064/ap113-1-1

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 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 $X$.

Authors

  • Shuzo IzumiResearch Center for Quantum Computing
    Kindai University
    Higashi-Osaka 577-8502, Japan
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image