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A+ CATEGORY SCIENTIFIC UNIT

Une note à propos du jacobien de fonctions holomorphes à l'origine de \mathbb{C}^n

Volume 94 / 2008

M. Hickel Annales Polonici Mathematici 94 (2008), 245-264 MSC: Primary 13A10; Secondary 13C10, 13J07. DOI: 10.4064/ap94-3-4

Abstract

Let f_{1},\ldots,f_{n} be n germs of holomorphic functions at the origin of \mathbb{C}^{n}, such that f_{i}(0)=0, 1\leq i\leq n . We give a proof based on J. Lipman's theory of residues via Hochschild homology that the jacobian of f_{1},\ldots,f_{n} belongs to the ideal generated by f_{1},\ldots,f_{n} if and only if the dimension of the germ of common zeros of f_{1},\ldots,f_{n} is strictly positive. In fact, we prove much more general results which are relative versions of this result replacing the field \mathbb{C} by convenient noetherian rings {\bf A} (Ths. 3.1 and 3.3). We then show a /Lojasiewicz inequality for the jacobian analogous to the classical one by S.~/Lojasiewicz for the gradient.

Authors

  • M. HickelÉquipe d'Analyse et Géométrie
    Université Bordeaux 1, I.M.B.
    and
    Département Informatique
    I.U.T. Bordeaux 1
    33405 Talence Cedex, France
    e-mail

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