Number fields in fibers: the geometrically abelian case with rational critical values

作者：Bilu Yuri^*; Luca Florian

来源：Periodica Mathematica Hungarica, 2017, 75(2): 315-321.

DOI：10.1007/s10998-017-0200-5

• Number fields
• Square-free numbers
• Polynomials
• Diversity conjectures

Let X be an algebraic curve over Q and t is an element of Q(X) a non-constant rational function such that Q(X) not equal Q(t). For every n is an element of Z pick P-n is an element of X((Q) over bar) such that t (P-n) = n. We conjecture that, for large N, among the number fields Q(P-1),..., Q(P-N) there are at least cN distinct. We prove this conjecture in the special case when (Q) over bar (X)/ (Q) over bar (t) is an abelian field extension and the critical values of t are all rational. This implies, in particular, that our conjecture follows from a more famous conjecture of Schinzel.

• 出版日期2017-12