On counting integral orbits and related applications

积分轨道计算及相关应用

• 批准号: RGPIN-2018-03975
• 负责人: Wang, XiaoHeng
• 金额: $ 1.68万
• 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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=679361
• 关键词: counting; integral; orbits; related; applications

One of the most fundamental questions in Number Theory is: given an equation (in possibly many variables) with integer coefficients, how many integer solutions does it have? For example, given a polynomial in one variable, one may ask how often does it take a square value. A general answer to this question will have a tremendous impact across the fields of Mathematics, Cryptography, and Computer Science. However, this question turns out to be extremely difficult. For example, we do not know how to answer it even when the polynomial has degree 3.******The question becomes somewhat more manageable if instead of considering only one equation, we consider an entire family of equations, and ask about the behavior of the number of solutions at random and on average. The study of solutions across a family of equations is called Arithmetic Statistics.******Instead of asking for polynomials taking square values, one may also ask for values that are as far from being squares as possible, numbers that are not a multiple of any square. Counting these numbers amounts to sieving out multiples of squares much like the classic sieve of Erastothenes for enumerating prime numbers, which lies at the heart of Analytic Number Theory.******My proposal is to tackle fundamental problems in both the fields of Arithmetic Statistics and Analytic Number Theory combining ideas from these fields and many other branches of mathematics including: Algebraic Geometry, a study of the set of solutions to polynomial equations in complex numbers; Arithmetic Invariant Theory, a passage from complex numbers to rational numbers and integers; and Analysis, or more precisely the method of geometry-of-numbers to count integer points in a region defined by inequalities. This will have a broad impact across Number Theory.

数论中最基本的问题之一是：给定一个具有整数系数的方程（可能有许多变量），它有多少整数解？例如，给定一个变量的多项式，人们可能会问它多久取一次平方值。这个问题的一般答案将对数学，密码学和计算机科学领域产生巨大影响。然而，这个问题变得非常困难。例如，即使多项式的次数为3，我们也不知道如何回答它。如果我们考虑整个方程族，而不是只考虑一个方程，并询问随机和平均解的数量的行为，那么这个问题就变得更容易处理了。研究一系列方程的解被称为算术统计。人们可以不求取平方值的多项式，而求取尽可能远离平方的值，即不是任何平方的倍数的数。计算这些数字相当于筛选出平方的倍数，就像埃拉斯托西尼的经典筛子用于枚举素数，这是解析数论的核心。我的建议是解决算术统计和解析数论领域的基本问题，结合这些领域和许多其他数学分支的思想，包括：代数几何，一组解决方案的研究多项式方程在复数;算术不变理论，从复数到有理数和整数的通道;和分析，或更准确地说，方法几何的数字来计算整数点在一个区域内定义的不等式。这将对数论产生广泛的影响。