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

Shuzo Izumi

Annales Polonici Mathematici, Tome 113 (2015), p. 1-42 / Harvested from The Polish Digital Mathematics Library

Résumé

Let (U) denote the algebra of holomorphic functions on an open subset U ⊂ ℂⁿ and Z ⊂ (U) its finite-dimensional vector subspace. By the theory of least spaces of de Boor and Ron, there exists a projection $_{b}$ from the local ring $_{n,b}$ 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 $_{b}$ 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 $_{b}$, 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 $_{X,a}$ 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.

Détail

Publié le : 2015-01-01

Zbl 1308.32011

EUDML-ID : urn:eudml:doc:286664