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 $\mathcal{F}$ be a differential field of characteristic 0 with $m$ derivation operators, $\text{u}=u_1,\dots ,u_n$ a set of differential indeterminates over $\mathcal{F}$. We prove that an intermediate differential field $\mathcal{G}$ between $\mathcal{F}$ and $\mathcal{F}⟨\text{u}⟩$ is a simple differential extension of $\mathcal{F}$ if and only if the differential dimension polynomial of $\text{u}$ over $\mathcal{G}$ is of the form ${\omega }_{\text{u}/\mathcal{G}}(t)=n\left(\genfrac{}{}{0ex}{}{t+m}{m}\right)-\left(\genfrac{}{}{0ex}{}{t+m-s}{m}\right)$ for some $s\in \mathbb{N}$. 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 $\mathcal{F}$ contained in $\mathcal{F}⟨\text{u}⟩$ 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.

19 pages