On the partial differential Lüroth's theorem

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