Effective differential Lüroth's theorem

Abstract

This paper focuses on effectivity aspects of the Lüroth's theorem in differential fields. Let F be an ordinary differential field of characteristic 0 and F〈u〉 be the field of differential rational functions generated by a single indeterminate u. Let be given non-constant rational functions v1,vn∈F〈u〉 generating a differential subfield G⊆F〈u〉. The differential Lüroth's theorem proved by Ritt in 1932 states that there exists v∈G such that G=F〈v〉. Here we prove that the total order and degree of a generator v are bounded by minjord(vj) and (n d(e +1) +1)2e +1, respectively, where e:=maxjord(vj) and d:=maxjdeg(vj). 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.