A differential Puiseux theorem in generalized series fields of finite rank

Matusinski, Mickaël

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. 2, pp. 247-293.

Nous étudions des équations différentielles $F(y,...,y^{\left(n\right)})=0$ où $F$ est une séries formelle en $y,y^{\prime },...,y^{\left(n\right)}$, à coefficients dans un corps de séries généralisées ${𝕂}_r$ de rang fini $r\in {\mathbb{N}}^{*}$. Notre objet est d’exprimer le support – c’est-à-dire l’ensemble $\mathrm{Supp}{y}_0$ des exposants – des éléments $y_0\in {𝕂}_r$ solutions, en fonction des supports des coefficients de l’équation, dont l’union est notée $\mathrm{Supp}F$.

We study differential equations $F(y,...,y^{\left(n\right)})=0$ where $F$ is a formal series in $y,y^{\prime },...,y^{\left(n\right)}$ with coefficients in some field of generalized power series ${𝕂}_r$ with finite rank $r\in {\mathbb{N}}^{*}$. Our purpose is to express the support $\mathrm{Supp}{y}_0$, i.e. the set of exponents, of the elements $y_0\in {𝕂}_r$ that are solutions, in terms of the supports of the coefficients of the equation, namely $\mathrm{Supp}F$.

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