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

M. Hickel¹

¹ Équipe d'Analyse et Géométrie Université Bordeaux 1, I.M.B. and Département Informatique I.U.T. Bordeaux 1 33405 Talence Cedex, France

Annales Polonici Mathematici, Tome 94 (2008) no. 3, pp. 245-264

Résumé

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.

Détail

DOI : 10.4064/ap94-3-4

Mots-clés : ldots germs holomorphic functions origin mathbb leq leq proof based lipmans theory residues via hochschild homology jacobian ldots belongs ideal generated ldots only dimension germ common zeros ldots strictly positive prove much general results which relative versions result replacing field mathbb convenient noetherian rings ths lojasiewicz inequality jacobian analogous classical lojasiewicz gradient