Effective Differential Lüroth's Theorem

Abstract

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 $\mathcal{G}\subseteq \mathcal{F}<eu>$. 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 ${\min }_j\text{ord}(v_j)$ and $(nd(e+1)+1{)}^{2e+1}$, respectively, where $e:={\max }_j\text{ord}(v_j)$ and $d:={\max }_j\text{deg}(v_j)$. 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.