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

TL;DR

This paper extends L"uroth's theorem to partial differential fields, providing a characterization of intermediate differential fields and an algorithm to identify simple extensions, with applications to differential curve re-parameterization.

Contribution

It generalizes the classical differential L"uroth's theorem to multiple derivations and introduces an algorithm for identifying simple differential extensions.

Findings

Characterization of intermediate differential fields via dimension polynomials

Algorithm to determine and compute L"uroth generators in differential extensions

Application to re-parameterization of unirational differential curves

Abstract

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.