有效微分Lüroth定理
Effective Differential Lüroth's Theorem
摘要：
本文主要讨论微分域中Lüroth定理的有效性问题。设$\mathcal{F}$是特征为0的常微分域，$\mathcal{F}<u>$是由单个不定$u$生成的微分有理函数域。给定生成微分子域$\mathcal{G}\subseteq \mathcal{F}<e u>$的非常数有理函数$v_1,...,v_n\in \mathcal{F}<u>$。Ritt在1932年证明的微分Lüroth定理指出存在$v\in \mathcal G$使得$\mathcal{G}= \mathcal{F}<v>$。本文证明了生成元$v$的全阶和全度分别由$\min_j \textrm{ord} (v_j)$和$(nd(e+1)+1)^{2e+1}$有界，其中$e:=\max_j \textrm{ord} (v_j)$和$d:=\max_j \textrm{deg} (v_j)$ .作为一个副产品，我们的技术使我们能够计算Lüroth生成器通过处理一个多项式环中的多项式理想在一个多变量。
英文摘要：
This paper focuses on effectivity aspects of the Lüroth's theorem in differential fields. Let $\mathcal{F}$ be an ordinary differential field of characteristic 0 and $\mathcal{F}<u>$ be the field of differential rational functions generated by a single indeterminate $u$. Let be given non constant rational functions $v_1,...,v_n\in \mathcal{F}<u>$ generating a differential subfield $\mathcal{G}\subseteq \mathcal{F}<e u>$. The differential Lüroth's theorem proved by Ritt in 1932 states that there exists $v\in \mathcal G$ such that $\mathcal{G}= \mathcal{F}<v>$. Here we prove that the total order and degree of a generator $v$ are bounded by $\min_j \textrm{ord} (v_j)$ and $(nd(e+1)+1)^{2e+1}$, respectively, where $e:=\max_j \textrm{ord} (v_j)$ and $d:=\max_j \textrm{deg} (v_j)$. As a byproduct, our techniques enable us to compute a Lüroth generator by dealing with a polynomial ideal in a polynomial ring in finitely many variables.
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