论文标题
在部分差异的Lüroth的定理上
On the Partial Differential Lüroth's Theorem
论文作者
论文摘要
我们研究了部分差分领域的吕伦问题。主要结果是以下广义lüroth定理的部分差分类似物：让$ \ Mathcal {f} $是特征的差异字段，具有$ m $衍生运算符，$ \ textbf {u} = u_1，\ ldots，\ ldots，\ ldots，u_n $一组差异$ $ \ mathcal} $ {f。我们证明，$ \ nathcal {f} $和$ \ nathcal {f} \ langle \ textbf {u} \ rangle $之间的中间差异字段$ \ MATHCAL {g} $是$ \ nomial dimial dymial dymial， $ \ MATHCAL {g} $是$ω_ {\ textbf {u}/\ Mathcal {g}}}（t）= n {t+m \ select m} - {t+m-s} $ in \ mathbb n $。该结果概括了Ritt和Kolchin在情况下证明了经典的差异Lüroth定理。然后，我们提出了一种算法，以确定给定有限生成的差异扩展字段$ \ mathcal {f} $是否包含在$ \ Mathcal {f} \ langle \ langle \ textbf {u} \ rangle $中是一个简单的扩展，并且在肯定的情况下，以计算LürothGenerator。作为应用程序，我们解决了针对Urirational差异曲线的适当重新参数化问题。
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, $\textbf{u}=u_1,\ldots,u_n$ 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 $ω_{\textbf{u}/\mathcal{G}}(t)=n{t+m\choose m}-{t+m-s\choose m}$ for some $s\in\mathbb N$. This result generalizes the classical differential 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 $\mathcal{F}\langle \textbf{u}\rangle$ 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.
