Dimension Growth for Affine Varieties

Abstract

Abstract We prove uniform upper bounds on the number of integral points of bounded height on affine varieties. If $X$ is an irreducible affine variety of degree $d\geq 4$ in ${\mathbb{A}}^{n}$, which is not the preimage of a curve under a linear map ${\mathbb{A}}^{n}\to{\mathbb{A}}^{n-\dim X+1}$, then we prove that $X$ has at most $O_{d,n,\varepsilon }(B^{\dim X - 1 + \varepsilon })$ integral points up to height $B$. This is a strong analogue of dimension growth for projective varieties, and improves upon a theorem due to Pila, and a theorem due to Browning–Heath-Brown–Salberger. Our techniques follow the $p$-adic determinant method, in the spirit of Heath-Brown, but with improvements due to Salberger, Walsh, and Castryck–Cluckers–Dittmann–Nguyen. The main difficulty is to count integral points on lines on an affine surface in ${\mathbb{A}}^{3}$, for which we develop point-counting results for curves in ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$. We also formulate and prove analogous results over global fields, following work by Paredes–Sasyk.