Formal relations between quasianalytic functions of some fixed class

F. Broglia¹ ; A. Elkhadiri² ; F. Pieroni¹

¹ Dipartimento di Matematica Via Filippo Buonarroti 2 56127 Pisa, Italy
² Département de Mathématiques et Informatique Faculté des Sciences Université Ibn Tofail B.P. 133 Kenitra, Morocco

Annales Polonici Mathematici, Tome 83 (2004) no. 1, pp. 35-40.

Résumé

In [Ga] Gabrielov has given conditions under which the completion of the kernel of a morphism $\varphi : A \to B$ between analytic rings coincides with the kernel of the induced morphism $\widehat {\varphi } : \widehat {A} \to \widehat {B}$ between the completions. If $B$ is a domain, a sufficient condition is that $\mathop {\rm rk}\nolimits \varphi = \mathop {\rm dim}\nolimits \! {(\widehat {A}/\! \mathop {\rm ker}\nolimits \widehat {\varphi })} $, where $\mathop {\rm rk}\nolimits \varphi $ is the rank of the jacobian matrix of $\varphi $ considered as a matrix over the quotient field of $B$. We prove that the above property holds in a fixed quasianalytic Denjoy–Carleman class if and only if the class coincides with the ring of germs of analytic functions.

Détail

DOI : 10.4064/ap83-1-4

Keywords: gabrielov has given conditions under which completion kernel morphism varphi between analytic rings coincides kernel induced morphism widehat varphi widehat widehat between completions domain sufficient condition mathop nolimits varphi mathop dim nolimits widehat mathop ker nolimits widehat varphi where mathop nolimits varphi rank jacobian matrix varphi considered matrix quotient field prove above property holds fixed quasianalytic denjoy carleman class only class coincides ring germs analytic functions