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

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Let 𝓞(U) denote the algebra of holomorphic functions on an open subset U ⊂ ℂⁿ and Z ⊂ 𝓞(U) its finite-dimensional vector subspace. By the theory of least spaces of de Boor and Ron, there exists a projection $𝖳_{b}$ from the local ring $𝓞_{n,b}$ onto the space $Z_{b}$ of germs of elements of Z at b. At a general point b ∈ U its kernel is an ideal and $𝖳_{b}$ induces the structure of an Artinian algebra on $Z_{b}$. In particular, this holds at points where the kth jets of elements of Z form a vector bundle for each k ∈ ℕ. For an embedded manifold $X ⊂ ℂ^{m}$, we introduce a space of higher order tangents following Bos and Calvi. In the case of curve, using $𝖳_{b}$, we define the Taylor projector of order d at a general point a ∈ X, generalising results of Bos and Calvi. It is a retraction of $𝓞_{X,a}$ onto the set of polynomial functions on $X_{a}$ 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 $X ⊂ ℂ^{m}$ is not very high at a general point of X.
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  • Research Center for Quantum Computing, Kindai University, Higashi-Osaka 577-8502, Japan
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