Dimension growth for affine varieties

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.