Counting rational points on affine hypersurfaces

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,