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


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 $X$.


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

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