Potential density, uniform boundedness, and points in special position

University Of Rochester, Rochester NY

Investigators

Abstract

This project focuses on the subject of dynamics and rational solutions to several polynomial equations. Polynomial equations over the rational numbers are among the oldest topics in all of mathematics; indeed, they are often called "Diophantine equations" after Diophantus of Alexandria, who was one of the first to study them, in the 3rd century AD. Many of the most important questions have been solved, at least partially, in the case of one and two dimensional systems of equations, but there is much that is unknown in dimensions three and up. One of the primary goals of this award is to use dynamics (simply the repeated application of a function) to solve an open question about solutions to certain types of three-dimensional systems of equations. Another goal is to develop a good notion of what it means for tuples of points to be in "dynamically special position", which means loosely that the points don't interact upon repeated application of a function. An important conjecture of Zhang asks that if f is a self-map of an algebraic variety X defined over a number field, then under reasonably general conditions, there should be an algebraic point of X whose orbit under f is Zariski dense in X. We seek to prove this conjecture in the case where f is an automorphism and X has dimension 3. The chief tool here is the p-adic parametrization of orbits, which may be thought of as a more general dynamical version of Skolem's method for treating linear recurrence sequences. Another goal of the project is to prove that if n is larger than 1, then given n distinct points in C and a list of n preperiods and periods, there is a polynomial of degree n+1 in normal form such that each point has the desired period and preperiod. The methods here are a combination of diophantine approximation over function fields and the "unlikely intersection" techniques pioneered by Masser, Zannier, Baker, DeMarco, and others.