Quantitative aspects of the generalized differential Lüroth's Theorem

D'ALFONSO, LISI; JERONIMO, GABRIELA; SOLERNÓ, PABLO

JOURNAL OF ALGEBRA

ACADEMIC PRESS INC ELSEVIER SCIENCE

Año: 2018 vol. 507 p. 547 - 570

0021-8693

Let F be a differential field of characteristic 0, t=t_1,...,t_m a finite set of differential indeterminates over F and G a subset of F<t> a differential field extension of F, generated by nonconstant rational functions alpha_1,...,alpha_n of total degree and order bounded by d and e>= 1 respectively. The generalized differential Lüroth´s Theorem states that if the differential transcendence degree of G over F is 1, there exists v in G such that G=F<v>.We prove a new explicit upper bound for the degree of v in terms of n, m, d and e. Further, we exhibit an effective procedure to compute v.