Year: 2025
Type: article
Abstract: Let $X$ be a smooth projective hypersurface defined over $\mathbb{Q}$. We provide new bounds for rational points of bounded height on $X$. In particular, we show that if $X$ is a smooth projective hypersurface in $\mathbb{P}^{n}$ with $n\geqslant 4$ and degree $d\geqslant 50$, then the set of rational points on $X$ of height bounded by $B$ have cardinality $O_{n,d,\varepsilon }(B^{n-2+\varepsilon })$. If $X$ is smooth and has degree $d\geqslant 6$, we improve the dimension growth conjecture bound. We achieve an analogue result for affine hypersurfaces whose projective closure is smooth.
DOI: 10.1093/imrn/rnaf249
Author: Matteo Verzobio
Institution: Institute of Science and Technology Austria
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $4,534
Funder: European Commission
OpenAlex ID: W4413186295