Spectrum of a linear differential equation over a field of formal power series

TL;DR

This paper introduces a new geometric spectrum for linear differential equations over Laurent formal power series fields, revealing insights into their structure through Berkovich's framework.

Contribution

It defines and computes a novel spectrum for such differential equations, linking algebraic and geometric perspectives.

Findings

Spectrum contains meaningful information about the differential equation

The spectrum is computed explicitly for the class of equations considered

Provides a new geometric tool for analyzing differential equations over formal power series

Abstract

In this paper we associate to a linear differential equation with coefficients in the field of Laurent formal power series a new geometric object, a spectrum in the sense of Berkovich. We will compute this spectrum and show that it contains interesting informations about the equation.