Potential density, uniform boundedness, and points in special position

这个项目的重点是动力学和几个多项式方程的有理解的主题。有理数上的多项式方程是所有数学中最古老的主题之一；事实上，它们通常被称为“丢番图方程”，以亚历山大的丢番图命名，他是公元3世纪最早研究它们的人之一。在一维和二维方程组的情况下，许多最重要的问题已经得到了解决，至少部分得到了解决，但在三维及三维以上的情况下，还有很多问题是未知的。该奖项的主要目标之一是使用动力学（简单地重复应用一个函数）来解决关于某些类型的三维方程组解的开放性问题。另一个目标是对点元组处于“动态特殊位置”意味着什么有一个很好的概念，这意味着点在重复应用函数时不会相互作用。Zhang的一个重要猜想是，如果f是定义在数域上的代数变量X的自映射，那么在合理的一般条件下，X上应该存在一个代数点，其在f下的轨道在X上是Zariski密的。我们寻求在f是自同构且X有维数为3的情况下证明这个猜想。这里的主要工具是轨道的p进参数化，它可以被认为是处理线性递归序列的Skolem方法的更一般的动力学版本。本课题的另一个目标是证明，如果n大于1，则给定C中n个不同的点和n个预周期和周期的列表，存在一个n+1次的范式多项式，使得每个点都有期望的周期和预周期。这里的方法结合了函数域上的丢芬图近似和由Masser、Zannier、Baker、DeMarco等人首创的“不可能相交”技术。

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.