Effective Differential Lüroth's Theorem

Abstract This paper focuses on effectivity aspects of the Luroth'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 v 1 , … , v n ∈ F 〈 u 〉 generating a differential subfield G ⊆ F 〈 u 〉 . The differential Luroth'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 min j ord ( v j ) and ( n d ( e + 1 ) + 1 ) 2 e + 1 , respectively, where e : = max j ord ( v j ) and d : = max j deg ( v j ) . As a byproduct, our techniques enable us to compute a Luroth generator by dealing with a polynomial ideal in a polynomial ring in finitely many variables.