Effective differential Luroth theorem

LISI D'ALFONSO; GABRIELA JERONIMO; PABLO SOLERNÓ

Nueva York

Conferencia; Applications of Computer Algebra; 2014

Let F be a differential field of characteristic 0 and F⟨u⟩ the field of differential rational functions in a single indeterminate u. The differential Luroth theorem proved by Ritt and extended by Kolchin states that for any differential subfield G of F⟨u⟩ there exists v such that G = F⟨v⟩. The talk will focus on effectivity aspects of this result. More precisely: given non-constant rational functions v1;...; vn  F⟨u⟩ such that G = F⟨v1;...; vn⟩, we will give upper bounds for the total order and degree of a Luroth generator v of the extension G/F in terms of the number and the maximum order and degree of the given generators of G. Our approach combines elements of Ritt´s and Kolchin´s proofs with estimations concerning the order and the differentiation index of differential ideals. As a byproduct, we will show that our techniques enable the computation of a Luroth generator by dealing with a polynomial ideal in a polynomial ring in finitely many variables.