### A differential Puiseux theorem in generalized series fields of finite rank

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${\mathbb{K}}_{r}$ with finite rank $r\in {\mathbb{N}}^{*}$. Our purpose is to express the support $\mathrm{Supp}\phantom{\rule{4pt}{0ex}}{y}_{0}$, i.e. the set of exponents, of the elements ${y}_{0}\in {\mathbb{K}}_{r}$ that are solutions, in terms of the supports of the coefficients of the equation, namely $\mathrm{Supp}\phantom{\rule{4pt}{0ex}}F$.