On the geometrization of a lemma of differential Galois theory

TL;DR

This paper geometrizes and generalizes a lemma from differential Galois theory, providing new insights and tools in differential algebra and algebraic geometry, including the study of simple D-schemes and D-rings.

Contribution

It introduces a geometric framework for a differential Galois lemma and extends the analysis to large families of vector fields and simple D-schemes.

Findings

Existence of coarse space of leaves for simple D-schemes

Development of the theory for large families of vector fields

New results on the existence of trajectories in differential algebra

Abstract

In this paper, we give a geometrization and a generalization of a lemma of differential Galois theory. This geometrization, in addition of giving a nice insight on this result, offers us the occasion to investigate several points of differential algebra and differential algebraic geometry. We study the class of simple \D-schemes and prove that they all have a coarse space of leaves. Furthermore, instead of considering schemes endowed with one vector field, we consider the case of arbitrarilly large families of vector fields. This leads us to some developments in differential algebra, in particular to prove the existence of the trajectory in this setting but also to study simple \D-rings.