| Résume | Given a projective algebraic variety X over Q, the dimension growth conjecture predicts general upper bounds for the number of points of bounded height on X. It was originally conjectured by Serre, and independently in a uniform way by Heath-Brown. By work of Browning, Heath-Brown and Salberger, uniform dimension growth is now a theorem.
I will give a general overview of dimension growth and explain some ideas of the proof. The main ingredient is the so-called determinant method, which goes back to Bombieri and Pila, and has been successfully applied to many counting problems. I will then turn to dimension growth for affine varieties, and report on recent work with Raf Cluckers, Pierre Dèbes, Yotam Hendel, and Kien Nguyen. |
