Potential density, uniform boundedness, and points in special position

这个项目的重点是动力学和合理的解决方案，以几个多项式方程的主题。 有理数上的多项式方程是所有数学中最古老的课题之一;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，但是在三维及以上的情况下，还有许多问题是未知的。 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）来解决某些类型的三维方程组的解决方案的开放问题。 另一个目标是发展一个很好的概念，它意味着什么元组的点是在“动态特殊位置”，这意味着松散的点不相互作用时，重复应用的功能。一个重要的猜想张问，如果f是一个自映射的代数簇X定义在数域上，那么在合理的一般条件下，应该存在X的一个代数点，其在f下的轨道在X中是Zebraki稠密的。 我们试图证明这一猜想的情况下，f是一个自同构和X的维数为3。 这里的主要工具是轨道的p-adic参数化，它可以被认为是Skolem处理线性递归序列的方法的更一般的动力学版本。 该项目的另一个目标是证明，如果n大于1，则给定C中的n个不同点和n个preperiod和period的列表，存在一个标准形式的n+1次多项式，使得每个点都有所需的周期和preperiod。这里的方法是函数域上的丢番图逼近和Masser，Zannier，Baker，德马科和其他人。

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.