Introduction

A now classical way of looking at systems, in the context of control, is to consider them as differential field extensions $K/k$. The concept of flat differential extension, introduced by Fliess et al. in [5,6], has produced many applications during the last decade. They are associated to systems linearizable by endogenous feedback, for which very efficient control methods have been developped.

DEFINITION 1. -- A flat extension $K/k$ is such that there exists an algebraic extension $K_{2}/K$ such that $K_{2}/k\langle z\rangle$ is an algebraic extension of a purely transcendental differential extension $k\langle y\rangle$.

The functions $z$ are called linearizing outputs.

We see that purely transcendental differential extensions are flat, but that not all flat extensions are purely transcendental.

If an algebraic curve $C\in k^{n}$ admits a rational parametrization $x\in C=f(y)$, it admits a unicursal one, i.e. a parametrization $x=g(z)$ such that a non singular point of the image admits at most one antecedent. Algebraically, this means that the field $k(f_{1}(y), \ldots, f_{m}(y))$ is equal to $k(z)$. This amounts to saying that any subextension $k\subset L\subset k(y)$ of a purely transcendental extension of transcendental degree $1$ is purely transcendental. That result is known as Lüroth theorem and it may be generalized to the case of surfaces, i.e. for transcendental degree $2$, provided that $k$ is algebraically closed.

Joseph F. Ritt has proved an analog of Lüroth theorem in the context of differential algebra (see [11]).

THEOREM 2. -- Any subextension $k\subset L\subset k\langle y\rangle$ of a purely transcendental differential extension of transcendental degree $1$ is a purely transcendental differential extension.