2022/10/11 by Wei Li, Li, Wei, Chen-Rui Wei +1
Computer Science · Mathematics · Medicine · #12H05 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #FOS: Computer and information sciences #FOS: Mathematics #Polynomial and algebraic computation #Spinal Hematomas and Complications #Symbolic Computation (cs.SC)
paper · pdf · doi:10.48550/arxiv.2210.05469
openalex publication_date 2022/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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=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)=nt+m\choose m-t+m-s\choose m for some s∈\mathbb 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.