Linear Differential Equation with Formal Power Series Non-Homogeneity Over a Ring with a Non-Archimedean Valuation

TL;DR

This paper studies linear differential equations with constant coefficients over a non-Archimedean valuation ring, providing conditions for solutions' existence and uniqueness, and constructing a fundamental solution in formal power series.

Contribution

It introduces sufficient conditions for the existence and uniqueness of solutions and constructs a fundamental solution in formal power series for such equations.

Findings

Established conditions for solution existence and uniqueness.

Constructed a fundamental solution in formal power series.

Showed convolution yields the unique solution.

Abstract

Consider the linear differential equation of $m$-th order with constant coefficients from the valuation ring $K$ of a non-Archimedean field. We get sufficient conditions of uniqueness and existence for the solution of this equation from $K[[x]]$. Also the fundamental solution from $\frac{1}{x}K[[\frac{1}{x}]]$ of the equation is obtained and it is shown that the convolution of the fundamental solution and a non-homogeneity is a unique solution of the equation.