On the geometrization of a lemma of Singer and van der Put
In this paper, we give a geometrization and a generalization of a lemma of differential Galois theory, used by Singer and van der Put in their reference book. This geometrization, in addition of giving a nice insight on this result, offers us the opportunity to investigate several points of differential algebra and differential algebraic geometry. We study the class of simple $\Delta$-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 arbitrarily large, and not necessarily commuting, 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 $\Delta$-rings.