On the Partial Differential Lüroth's Theorem
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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 derivation operators, u=u_1,…,u_n 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 ω_u/𝒢(t)=nt+m m-t+m-s m 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.
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