Counting rational points on affine hypersurfaces

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.