The Morton–Silverman Conjecture

Uniform boundedness for preperiodic points: fix integers \(N \geq 1\), \(d \geq 2\) and \(D \geq 1\). There is a bound \(C(N, d, D)\), depending only on \(N\), \(d\) and \(D\), such that no morphism \(\varphi: \mathbb{P}^N \to \mathbb{P}^N\) of degree \(d\) defined over a number field \(K\) with \([K:\mathbb{Q}] \leq D\) has more than \(C(N, d, D)\) preperiodic points in \(\mathbb{P}^N(K)\).

Conjectured by Patrick Morton and Joseph Silverman in 1994 as the dynamical analogue of Merel's uniform boundedness for torsion on elliptic curves (itself conjectured by Manin and proved by Merel in 1996). Small cases are known — quadratic polynomials over \(\mathbb{Q}\) admit at most a handful of rational preperiodic points (Poonen 1998; Stoll, Hutz) — but no bound uniform in the map is known in any degree \(\geq 2\), and the conjecture drives much of modern arithmetic dynamics.