Polynomials with many rational preperiodic points

How many rational preperiodic points can a degree d polynomial in Q[x] have? Conjecturally, there is some uniform bound B_d(Q) on the number of such preperiodic points — but how big is B_d(Q)? Using interpolation we can easily find examples with d + 1 rational preperiodic points, but every point beyond that has to be fought for. In this talk I will share some recent work, joint with John R. Doyle, in which we prove that B_d(Q) is at least d + c*log(d) for some constant c and for all sufficiently large d. As a bonus, I’ll show how our construction also produces examples of pairs of degree d polynomials f(x) and g(x) with more than d^2 common complex preperiodic points.

http://people.math.harvard.edu/~demarco/AlgebraicDynamics/ for more information