Diophantine geometry and arithmetic dynamics

丢番图几何与算术动力学

• 批准号: 371987-2009
• 负责人: Ingram, Patrick
• 金额: $ 1.31万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2009
• 资助国家: 加拿大
• 起止时间: 2009-01-01 至 2010-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=435050
• 关键词: Diophantine; geometry; arithmetic; dynamics

My research in number theory deals largely with problems coming from dynamical systems. In particular, my research deals with questions of the following form: given a rational function and some initial value (a rational number), what are the arithmetic properties of the sequence of numbers generated by repeatedly applying the function to the initial value. For example, if a sequence of this form falls into a repeating cycle, one would like to know how long such a cycle might be, given some information about the function. One the other hand, if the sequence never repeats, then the "heights" of the terms (essentially the number of digits in numerator and denominator of these rational numbers) grow at an exponential rate. One significant problem in the area, one which I have made some progress on, and plan to research further, is to obtain explicit lower bounds on the exact rate of this exponential growth, for certain families of rational functions. This research has implications in various areas of number theory (to elliptic curves, recurrence sequences, etc.), as well as to problems in dynamics, computation, and mathematical logic.

我在数论方面的研究主要涉及来自动力系统的问题。特别是，我的研究涉及以下形式的问题：给定一个有理函数和某个初值(有理数)，通过将该函数重复应用于初值而生成的数字序列的算术性质是什么。例如，如果这种形式的序列落入一个重复的循环，人们想知道这样的循环可能有多长，给定一些关于该函数的信息。另一方面，如果这个序列永远不会重复，那么这些项的“高度”(本质上就是这些有理数的分子和分母中的位数)会以指数速度增长。这个领域中的一个重要问题，也是我已经取得一些进展并计划进一步研究的问题，是对于某些有理函数族，获得这种指数增长率的确切下界的显式下界。这项研究涉及数论的各个领域(椭圆曲线、递归序列等)，以及动力学、计算和数理逻辑中的问题。