Algorithmic Differential Lüroth Theorem

LISI D'ALFONSO; GABRIELA JERONIMO; PABLO SOLERNÓ

Córdoba

Congreso; IV Congreso Latinoamericano de Matemáticos; 2012

Let F be a differential field of characteristic 0 and F the field of differential rational functions in a single indeterminate u. The differential Lüroth theorem proved by Ritt states that for any differential subfield G of F there exists v in G such that G = F<v>. The talk will focus on effectivity aspects of this result. More precisely, we will address the following algorithmic problem: given rational functions v_1,...,v_n in F such that G = F<v_1,...,v_n>, compute a Lüroth generator v of the extension G/F. Our approach combines elements of Ritt´s and Kolchin´s proofs with estimations concerning the order and the differentiation index of differential ideals. We prove upper bounds for the total order and degree of a Lüroth generator v in terms of the number and the maximum order and degree of the given generators of G. These estimations allow us to reduce the problem to computations in a polynomial ring in finitely many variables.