Rational and integral values of rational functions at rational points
Abstract
The basic issue concerns sets of values of rational functions at rational points of an algebraic variety, namely image f(X(k)), where X is an algebbaric variety and f is a rational function on X, defined over the number field k. For instance, we shall prove that for X an abelian variety, the map between rational points is never surjective. This is reminiscent of the Hilbert Property, but here the fibers may have arbitrary dimension. One of our examples concerns the classical Hilbert Property: we produce a simply connected affine surface whose set of integral points is Zariski-dense and thin, disproving a plausible expectation. We shall also discuss hieghts and integrality issues; in this context, a role will be played by 'gcd estimates'. In the first Appendix, written by D. Masser, an effective estimate of some relevant gcd is provided.
Related
- Principle of generating critical points of rational functions and Gauss-Fourier conjecture for second-kind zeros
- Potential density of rational points on sextic double solids
- Rational Points on Danielewski Surfaces over Finite fields and Polygonal Numbers
- Rational points on loxodromic K3 surfaces and complex dynamics
- Large values of the Hurwitz zeta function with rational parameter
BibTeX
@misc{corvaja2026rational,
title = {{Rational and integral values of rational functions at rational points}},
author = {Pietro Corvaja and Umberto Zannier and appendix by D. Masser},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28255},
eprint = {2608.28255},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28255},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rational and integral values of rational functions at rational points
AU - Pietro Corvaja
AU - Umberto Zannier
AU - appendix by D. Masser
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28255
UR - https://arxiv.org/abs/2608.28255
AB - The basic issue concerns sets of values of rational functions at rational points of an algebraic variety, namely image f(X(k)), where X is an algebbaric variety and f is a rational function on X, defined over the number field k. For instance, we shall prove that for X an abelian variety, the map between rational points is never surjective. This is reminiscent of the Hilbert Property, but here the fibers may have arbitrary dimension. One of our examples concerns the classical Hilbert Property: we produce a simply connected affine surface whose set of integral points is Zariski-dense and thin, disproving a plausible expectation. We shall also discuss hieghts and integrality issues; in this context, a role will be played by 'gcd estimates'. In the first Appendix, written by D. Masser, an effective estimate of some relevant gcd is provided.
ER -