Effective Differential L\"uroth's Theorem

TL;DR

This paper establishes effective bounds on the order and degree of generators in differential L"uroth's theorem within differential fields, and provides a method to compute such generators using polynomial ideals.

Contribution

It introduces explicit bounds for the order and degree of generators in differential L"uroth's theorem and offers a computational approach to find these generators.

Findings

Bounds on the order and degree of generators are established.

A polynomial ideal approach enables computation of L"uroth generators.

The results improve understanding of effectivity in differential algebra.

Abstract

This paper focuses on effectivity aspects of the L\"uroth'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 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 $\min_j \textrm{ord} (v_j)$ 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 L\"uroth generator by dealing with a polynomial ideal in a polynomial ring in finitely many variables.