Counting rational points on affine hypersurfaces

TL;DR

This paper establishes an upper bound on the number of rational points of bounded height on irreducible affine hypersurfaces, advancing understanding of rational solutions in algebraic geometry.

Contribution

It introduces a quantitative form of Hilbert's irreducibility theorem to bound reducible specialisations, leading to new bounds on rational points on hypersurfaces.

Findings

Proved an explicit upper bound for rational points on hypersurfaces.

Developed a quantitative Hilbert irreducibility theorem.

Bounded reducible specialisations at rational points.

Abstract

We prove an upper bound for the number of rational points of bounded height on irreducible affine hypersurfaces. More precisely, given an irreducible polynomial $f \in \mathbb{Z}[X_1, \dots, X_n]$, we prove an upper bound on the number of points $(x_1, \dots, x_n) \in \mathbb{Q}^n$ such that $f(x_1, \dots, x_n) = 0$ and each component has height at most $B$. To prove this, we require a quantitative form of Hilbert's irreducibility theorem, where we bound the number of reducible specialisations of an irreducible polynomial at rational points of bounded height.