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## Formal relations between quasianalytic functions of some fixed class

### Tom 83 / 2004

Annales Polonici Mathematici 83 (2004), 35-40 MSC: Primary 32B20. DOI: 10.4064/ap83-1-4

#### Streszczenie

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.

#### Autorzy

Via Filippo Buonarroti 2
56127 Pisa, Italy
e-mail
• A. ElkhadiriDépartement de Mathématiques et Informatique
Faculté des Sciences
Université Ibn Tofail
B.P. 133
Kenitra, Morocco
e-mail
• F. PieroniDipartimento di Matematica
Via Filippo Buonarroti 2
56127 Pisa, Italy
e-mail

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