Effective Differential Lüroth's Theorem

Details Effective Differential Lüroth's Theorem Effective Differential Lüroth's Theorem

2012-02-28

arxiv.org/abs/1202.6344v1

Mathematics

Commutative Algebra

Scientific paper

This paper focuses on effectivity aspects of the L\"uroth's theorem in differential fields. Let $\mathcal{F}$ be a 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 subfield $\mathcal{G}\subseteq \mathcal{F}$. The differential L\"uroth'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 $\min_j \text{ord} (v_j)$ and $(nd(e+1)+1)^{2e+1}$, respectively, where $e:=\max_j \text{ord} (v_j)$ and $d:=\max_j \text{deg} (v_j)$. We also present a new probabilistic algorithm which computes the generator $v$ with controlled complexity.