Integral points, divisibility between values of polynomials and entire curves on surfaces

TL;DR

This paper establishes new degeneracy results for integral points and entire curves on surfaces, including the first example of a simply connected smooth variety with non-Zariski-dense integral points and entire curves.

Contribution

It introduces novel degeneracy results and provides the first example of a simply connected smooth variety with non-Zariski-dense integral points and entire curves.

Findings

Integral points on certain surfaces are not Zariski-dense.

Constructed the first example of a simply connected smooth variety with this property.

Connected results to divisibility problems involving polynomial values.

Abstract

We prove some new degeneracy results for integral points and entire curves on surfaces; in particular, we provide the first example, to our knowledge, of a simply connected smooth variety whose sets of integral points are never Zariski-dense (and no entire curve has Zariski-dense image). Some of our results are connected with divisibility problems, i.e. the problem of describing the integral points in the plane where the values of some given polynomials in two variables divide the values of other given polynomials.