Differential Puiseux theorem in generalized series fields of finite rank

TL;DR

This paper investigates differential equations over generalized power series fields of finite rank, exploring how the exponents of solutions relate to the exponents of the equations' coefficients.

Contribution

It extends the differential Puiseux theorem to generalized series fields of finite rank, establishing a connection between the supports of solutions and coefficients.

Findings

Established a generalized differential Puiseux theorem for finite rank series fields

Characterized the support of solutions in relation to the support of the equations

Provided new insights into the structure of solutions in generalized series fields

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 $r\in\mathbb{N}^*$. Our purpose is to understand the connection between the set of exponents of the coefficients of the equation $\textrm{Supp} F$ and the set $\textrm{Supp} y_0$ of exponents of the elements $y_0\in\mathds{K}_r$ that are solutions.