On the locus Per_n(lambda) of degree-d polynomials carrying a period-n point of multiplier lambda, no single critical point can account for nearly all of the critical height: every such map either has moduli height bounded linearly in the height of lambda, or has two independent critical points whose canonical heights are each a fixed positive fraction of its height. Students learn how a purely local argument — at each absolute value, once the critical points are large some branch point is large enough to escape to infinity — is summed over the places that carry a positive proportion of the height to yield a global statement. The consequence is that polynomials with a parabolic fixed point and fewer than two independent infinite critical orbits form a set of bounded height, hence a finite set over any fixed number field, with function-field analogues characterizing isotriviality and a matching bound for quadratic rational maps.
Degree-d polynomial maps are classified up to change of coordinates by points of a moduli space P_d, on which a Weil height h measures arithmetic complexity, while each map f carries a canonical heig…