# A Note on Algebraic Linear Partial Differential Equations

## TL;DR

This paper demonstrates that a broad class of algebraic linear partial differential equations and ordinary differential equations have trivial solutions, using jet-module techniques and base change theory, highlighting differences from the differentiable case.

## Contribution

It introduces a general algebraic framework showing zero kernel for broad classes of algebraic PDEs and ODEs, contrasting with the differentiable case.

## Key findings

- Algebraic linear PDE systems have zero kernel.
- Algebraic ODE systems also have zero kernel.
- Techniques involve jet-modules and base change theory.

## Abstract

In this note, we show that a very general system of algebraic linear partial differential equations has zero kernel, applying basic techniques of the theory of jet-modules and elementary base change theory. In particular, in contrast to the differentiable case, a very general system of algebraic ordinary differential equations has zero kernel.