Jonathan Pila

The number of integral points on arcs and ovals

integral lattice points, and that the exponent and constant are best possible. However, Swinnerton–Dyer [10] showed that the preceding result can be substantially improved if we start with a fixed, C, strictly convex arc Γ and consider the number of lattice points on tΓ, the dilation of Γ by a factor t, t ≥ 1. This of course is the same as asking for rational points (mN , n N ) on Γ as N → ∞. In fact, Swinnerton–Dyer proves a bound of type |tΓ ∩ ZZ| ≤ c(Γ, e)t 3 5+e

The rational points of a definable set

Let

Rational points in periodic analytic sets and the Manin–Mumford conjecture

X\R^n

Geometric postulation of a smooth function and the number of rational points

be a set that is definable in an o-minimal structure over

Some unlikely intersections beyond André–Oort

R

Detecting perfect powers by factoring into coprimes

. This article shows that in a suitable sense, there are very few rational points of

Ax–Schanuel for the

X

j

which do not lie on some connected semialgebraic subset of

-function

X

A correction to “Counting rational points on a certain exponential-algebraic surface”@@@Une correction au « Counting rational points on a certain exponential-algebraic surface »

of positive dimension