Number Theory Seminar

Thursday 3/02/06 4:00-5:00 PM in Hylan 1101:
Rob Benedetto-Towards Uniform Boundedness for Polynomial Dynamics

ABSTRACT:

Let K be a global field, and let f(z) in K(z) be a rational function defined over K of degree at least two. Clearly f maps K-rational points to K-rational points. In 1950, Northcott proved that f has only finitely many K-rational preperiodic points. In fact, Northcott's result holds for morphisms of P^N for arbitrary dimension N. In 1994, Morton and Silverman formulated a broad conjecture stating that the number of K-rational preperiodic points in K is bounded by a constant depending only on K, the degree of f, and the dimension N. In this talk, we will present improved (but still non-uniform) bounds for the number of K-rational preperiodic points in dimension one in the case that f is a polynomial. If time permits, we will also discuss the current state of knowledge for function fields of characteristic zero.