Main theorem

Ritt asked whether a two-dimensional generalization exists in the differential case as it does in the algebraic one. A slight modification of an example of a flat system due to Pierre Rouchon was enough to produce a counter-example to a problem which was considered «entirely outside the techniques so far available» (A. Buium [1] p. 111). See [9] for the counter-example. So, we cannot expect to generalize Lüroth theorem to differential fields outside the one-dimensional case without weakening the conclusions of the theorem.

We will prove the following version.

THEOREM 3. -- If $k\subset L\subset K$ is a subextension of a flat extension $K/k$, the extension $L/k$ is flat.

The control-theoretic analog of this theorem is that a system linearizable by dynamic feedback is linearizable by endogenous feedback.