Let f be an endomorphism of Pm of degree d defined over a number field K of degree n, and let H denote the absolute multiplicative height. Recently, we proved that the number of points P in Pm(K) with H(f(P)) < X is asymptotically equal to an explicit constant cK(f) times Xn(m+1)/d plus a power-saving error term. (When f is the identity map, we recover Schanuel’s theorem.) The constant cK(f) depends on the classical invariants of K, the height of f, and the behaviour of f at all its places of bad reduction.

In this talk, we will sketch the proof of the above formula, with an emphasis on the decomposition of the constant cK(f) in terms of local factors. Time permitting, we will discuss two applications: (1) counting points in the image of a morphism, and (2) counting points by canonical height.