On the rings of formal solutions of polynomial differential equations
| Autor: | Maria-Angeles Zurro |
|---|---|
| Rok vydání: | 1998 |
| Předmět: |
Noetherian
Pure mathematics Artin approximation theorem Partial differential equation Series (mathematics) Ordinary differential equation Mathematical analysis Structure (category theory) General Earth and Planetary Sciences Order (ring theory) Type (model theory) General Environmental Science Mathematics |
| Zdroj: | Banach Center Publications. 44:277-292 |
| ISSN: | 1730-6299 0137-6934 |
| DOI: | 10.4064/-44-1-277-292 |
| Popis: | Introduction. The Gevrey series appeared as formal solutions of partial differential equations of second order ([G]). In 1903 Maillet ([Mi]) proved that the formal solutions of ordinary differential equations with polynomial coefficients are of this type. Afterwards, Malgrange ([Ml]) and J. Cano ([Ca]) generalized this result to ordinary differential equations with analytic and Gevrey coefficients respectively. But for partial differential equations the analogous result is not yet achieved for any partial differential equation with polynomial coefficients; nevertheless there are some important results: see Ouchi ([O]). The semianalytic geometry with Gevrey conditions to the border is studied in the article of Tougeron [T]. He generalized the basic theorems of the semianalytic geometry to this case. As far as we know, a study of the algebraic properties of this series has not yet been done. This work shows the basic algebraic properties of these rings. In particular, they are noetherian and henselian. The basic tools we use here are the pseudo-Banach structure they have, and the formal Borel transform. This transform changes the Gevrey series by |
| Databáze: | OpenAIRE |
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