Differential Puiseux theorem in generalized series fields of finite rank

Abstract

We study differential equations $F(y,...,y^{(n)})=0$ where $F(Y_0,...,Y_n)$ is a formal series in $Y_0,...,Y_n$ with coefficients in some field of \emph{generalized power series} $\mathds{K}_r$ with finite rank $r\in {\mathbb{N}}^{\ast }$. Our purpose is to understand the connection between the set of exponents of the coefficients of the equation $\text{Supp}F$ and the set $\text{Supp}{y}_0$ of exponents of the elements $y_0\in\mathds{K}_r$ that are solutions.

arXiv comment37 pages