Algebraic dynamics

代数动力学

• 批准号: 0801072
• 负责人: Thomas Tucker
• 金额: $ 14.99万
• 依托单位: University of Rochester
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801072&HistoricalAwards=false
• 关键词: Algebraic; dynamics

This project focuses on the subject of algebraic dynamics. Orbits ofpoints under iterates of maps are among the most important objects indynamics. In the study of algebraic dynamics, the map is typically apolynomial or rational map in one or more variables. It is natural toask what kinds of algebraic relations infinite families of elementswithin an orbit can satisfy. The PI conjectures that if infinitelymany elements within an orbit satisfy an algebraic relation, then thatrelation must in some sense be stable under some iterate of the map.This conjecture may be thought of as a general dynamical analog of thewell-known Mordell-Lang conjecture for group varieties. Thus far,this conjecture has been proved in a few special cases using acombination of diophantine and p-adic analytic methods. The main goalof this project is to prove the conjecture in general. This project focuses on the interaction between algebraic maps andalgebraic equations. An algebraic map is a function such as f(x) =2x+3. Applying the map repeatedly to a single number gives what iscalled the orbit of that number under f. For example, if f(x) = 2x+3and we start with the number 1, then the orbit is 1, 5, 13, 29, 61,and so on. One can also form orbits out of pairs, triplets, andn-tuples of numbers by allowing f to be an algebraic map in more thanone variable. An algebraic equation is simply a more general versionof the familiar quadratic polynomials one encounters in beginningalgebra, except that it may have more variables and have higher degreeterms. One expects that if infinitely many numbers in an orbitsatisfy some algebraic relation, then that relation has an innateconnection with the algebraic map. The PI states a precise conjecturealong these lines, which he hopes to prove. The strategy for theproof involves a wide range of techniques, some number theoretic, somegeometric, and some analytic.

这个项目的重点是代数动力学的主题。 点在映射迭代下的轨道是动力学中最重要的对象之一。 在代数动力学的研究中，映射通常是关于一个或多个变量的多项式或有理映射。 很自然地要问什么样的代数关系无限族元素在一个轨道可以满足。 PI证明了如果一个轨道中有无穷多个元素满足一个代数关系，那么这个关系在某种意义上在映射的某个顶点下一定是稳定的，这个猜想可以被认为是著名的群簇的Mordell-Lang猜想的一般动力学模拟。 到目前为止，这一猜想已在几种特殊情况下用丢番图和p-adic分析方法得到了证明。 这个项目的主要目标是从总体上证明这个猜想。这个项目的重点是代数地图和代数方程之间的相互作用。代数映射是一个函数，如f（x）=2x+3。 将这个映射反复应用于一个数，就给出了这个数在f下的轨道。 例如，如果f（x）= 2x+3，我们从数字1开始，那么轨道是1，5，13，29，61，等等。我们也可以通过允许f是一个多变量的代数映射来形成由数对，三元组和n元组组成的轨道。 一个代数方程只是一个更一般的版本熟悉的二次多项式一遇到代数，除了它可能有更多的变量和更高的degreetries。 人们期望，如果一个轨道中的无穷多个数满足某种代数关系，那么这种关系与代数映射有着内在的联系。 PI沿着这些路线陈述了一个精确的假设，他希望证明这一点。 证明的策略涉及到广泛的技术，一些数论的，一些几何的，还有一些分析的。