Dimension Growth for Affine Varieties

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 4 in A^n, which is not the preimage of a curve under a linear map A^nA^n- X+1, then we prove that X has at most O₃, ₍, (B^ X - 1 +) 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 A^3, for which we develop point-counting results for curves in P^1P^1. We also formulate and prove analogous results over global fields, following work by Paredes–Sasyk.