Intersections of Polynomial Orbits, and a Dynamical Mordell-Lang Conjecture

MEMBERS SEMINAR
Topic:Intersections of Polynomial Orbits, and a Dynamical Mordell-Lang Conjecture
Speaker:Michael Zieve
Affiliation:Member, School of Mathematics
Date:Wednesday, April 13
Time/Room:2:00pm - 3:00pm/S-101

Let f and g be nonlinear polynomials (in one variable) over the complex numbers. I will show that, if there exist complex numbers a and b for which the orbits {a, f(a), f(f(a)), ...} and {b, g(b), g(g(b)), ...} have infinite intersection, then f and g have a common iterate (i.e., some f(f(f(...(f(x))...))) = g(g(...(g(x))...))). The proof involves Siegel's theorem on integral points on curves, results on factors of "variables separated" polynomials f(x)-g(y), and solutions to functional equations in Laurent polynomials. I will then explain a general problem which simultaneously generalizes both this result and the Mordell conjecture. I promise to make this talk accessible to absolutely everybody in the School of Mathematics!