Effective differential Lüroth's theorem

LISI D'ALFONSO; GABRIELA JERONIMO; PABLO SOLERNÓ

JOURNAL OF ALGEBRA

ACADEMIC PRESS INC ELSEVIER SCIENCE

Año: 2014 vol. 406 p. 1 - 19

0021-8693

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 v_1,...,v_n in F<u> generating a differential subfield G of F<u>. The differential Lüroth´s theorem proved by Ritt in 1932 states that there exists v in 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 (nd(e+1)+1)^{2e+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 Lüroth generator by dealing with a polynomial ideal in a polynomial ring in finitely many variables.