Effective Differential Lüroth's Theorem

This paper focuses on effectivity aspects of the Lüroth’s theorem in differential fields. Let F be a 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 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 minj ord(vj) and (nd(e + 1) + 1) , respectively, where e := maxj ord(vj) and d := maxj deg(vj). We also present a new probabilistic algorithm which computes the generator v with controlled complexity.