On the partial differential Lüroth's theorem

Abstract

We study the Lüroth problem for partial differential fields. The main result is the following partial differential analog of generalized Lüroth's theorem: Let F be a differential field of characteristic 0 with m commuting derivation operators, u=u1,…,un a set of differential indeterminates over F. We prove that an intermediate differential field G between F and F〈u〉 is a simple differential extension of F if and only if the differential dimension polynomial of u over G is of the form ωu/G(t)=n(t+mm)−(t+m−sm) for some s∈N. This result generalizes the classical differential Lüroth's theorem proved by Ritt and Kolchin in the case m=n=1. We then present an algorithm to decide whether a given finitely generated differential extension field of F contained in F〈u〉 is a simple extension, and in the affirmative case, to compute a Lüroth generator. As an application, we solve the proper re-parameterization problem for unirational differential curves.