On the Partial Differential L\"{u}roth's Theorem

We study the L\"{u}roth problem for partial differential fields. The main result is the following partial differential analog of generalized L\"{u}roth's theorem: Let $\mathcal{F}$ be a differential field of characteristic 0 with $m$ derivation operators, $\textbf{u}=u_1,\ldots,u_n$ a set of differential indeterminates over $\mathcal{F}$. We prove that an intermediate differential field $\mathcal{G}$ between $\mathcal{F}$ and $\mathcal{F}\langle \textbf{u}\rangle$ is a simple differential extension of $\mathcal{F}$ if and only if the differential dimension polynomial of $\textbf{u}$ over $\mathcal{G}$ is of the form $\omega_{\textbf{u}/\mathcal{G}}(t)=n{t+m\choose m}-{t+m-s\choose m}$ for some $s\in\mathbb N$. This result generalizes the classical differential L\"uroth'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 $\mathcal{F}\langle \textbf{u}\rangle$ is a simple extension, and in the affirmative case, to compute a L\"{u}roth generator. As an application, we solve the proper re-parameterization problem for unirational differential curves.