Uniformity of Rational Points

Lang's conjectures, conjecture H, and uniformity

We prove that Conjecture H of Caporaso et al. [CHarM], §6 together with Lang’s conjecture implies the uniformity of rational points on varieties of general type, as predicted in [CHarM]; a few applications in arithmetic and geometry are stated. As in [א], one uses the fact that rational points on X B are n-tuples of points when trying to bound n, as well as the fact that the Noetherian inductio...

Distribution of Rational Points: A Survey

Lang’s Conjectures, Fibered Powers, and Uniformity

We prove that the fibered power conjecture of Caporaso et al. (Conjecture H, [CHM], §6) together with Lang’s conjecture implies the uniformity of rational points on varieties of general type, as predicted in [CHM]; a few applications on the arithmetic and geometry of curves are stated. In an opposite direction, we give counterexamples to some analogous results in positive characteristic. We sho...

Points at Rational Distance from the Vertices of a Unit Polygon

A circuit uniformity sharper than DLogTime

We consider a new notion of circuit uniformity based on the concept of rational relations, called Rational-uniformity and denoted Rat. Our goal is to prove it is sharper than DLogTime-uniformity, the notion introduced by Barrington et al. [3], denoted DLT , that is: 1) Rational-uniformity implies DLogTime-uniformity, 2) we have NCRat NCDLT ( NCRat, 3) we have ∀k > 0, NCDLT ⊆ NC Rat , 4) Rationa...