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On the Łojasiewicz exponent of the gradient of a holomorphic function

Andrzej Lenarcik (1998)

Banach Center Publications

The Łojasiewicz exponent of the gradient of a convergent power series h(X,Y) with complex coefficients is the greatest lower bound of the set of λ > 0 such that the inequality | g r a d h ( x , y ) | ≥ c | ( x , y ) | λ holds near 0 ∈ C 2 for a certain c > 0. In the paper, we give an estimate of the Łojasiewicz exponent of grad h using information from the Newton diagram of h. We obtain the exact value of the exponent for non-degenerate series.

Ouverts analytiques d'une courbe algébrique en géométrie rigide

Qing Liu (1987)

Annales de l'institut Fourier

Nous étudions les espaces analytiques rigides de dimension 1, réguliers, de genre fini sur un corps valué complet k . Nous montrons qu’un tel espace X admet une réduction préstable. Si k est maximalement complet, X se plonge dans une courbe algébrique (analytifiée). On donne aussi une caractérisation des espaces analytiques qui sont le complémentaire d’une partie compacte dans une courbe algébrique.

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

Shuzo Izumi (2015)

Annales Polonici Mathematici

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...

The Łojasiewicz exponent of c-holomorphic mappings

Maciej P. Denkowski (2005)

Annales Polonici Mathematici

The aim of this paper is to study the Łojasiewicz exponent of c-holomorphic mappings. After introducing an order of flatness for c-holomorphic mappings we give an estimate of the Łojasiewicz exponent in the case of isolated zero, which is a generalization of the one given by Płoski and earlier by Chądzyński for two variables.

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