Number Fields in Fibers: the Geometrically Abelian Case with Rational Critical Values

Let X be an algebraic curve over Q and t a non-constant Q-rational function on X such that Q(t) is a proper subfield of Q(X). For every integer n pick a point P_n on X 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 t defines a geometrically abelian covering of the projective line, and the critical values of t are all rational. This implies, in particular, that our conjecture follows from a famous conjecture of Schinzel.Comment: Some typos are corrected. The article is now accepted in Periodica Math. Hungaric

The (α+2β)-Inequality on a Torus

A theorem of Macbeath asserts that μ(A+B)≥min(1, μ(A)+μ(B)) for any subsets A and B of a finite-dimensional torus. We conjecture that, when the obvious exceptions are excluded, a stronger inequality μ¯(A+B)≥min(1,μ¯(A)+μ¯(B)+min(μ¯(A),μ¯(B))) holds, and we prove this conjecture under some technical restriction

Quantitative Riemann existence theorem over a number field

Given a covering of the projective line with ramifications defined over a number field, we define a plain model of the algebraic curve realizing the Riemann existence theorem for this covering, and bound explicitly the defining equation of this curve and its definition field.Comment: 23 pages, version 4, minor change