Members’ Seminar

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!