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.
Categories: math.AG cs.SC