Finite morphisms of differential algebraic varieties and elimination theory

The main result of this Note is the following: for an algebraic system which evolution depends on variables which are partitionned into w and z , the elimination of the z leads to one set of differential algebraic equations (hence, with no inequations ) if the projection map along z is a finite morphism of algebraic varieties; that is, if the differential algebra which defines the system is integral over a suitable differential subalgebra. To obtain this result, is lifted to differential algebra a more general, and well-known result in algebraic geometry which states that a finite morphism of algebraic varieties is a closed one with respect to the Zariski topology.