Algebraic independence and Diophantine approximation

代数独立性和丢番图近似

• 批准号: RGPIN-2014-05086
• 负责人: Roy, Damien
• 金额: $ 1.68万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=588203
• 关键词: Algebraic; independence; Diophantine; approximation

My research programme addresses two fundamental problems in number theory. The first one deals with algebraic independence of values of the usual exponential function. The Graal here is a very general conjecture of Schanuel which contains all known results and all generally accepted conjectures on these values. For example it implies the transcendence of the number pi, proved by Lindemann in 1882, which in turn shows the impossibility of the ancient Greek problem of squaring the circle. In 2001, I proved that a construction of auxiliary function due to M. Waldschmidt should suffice to attack this conjecture. This seems to be the first realistic programme towards solving the conjecture. It reduces the problem to what I call a “small value estimate”, the problem of analyzing when a polynomial can take small values at many points of a highly structured set. It calls for a result that would encompass the wonderful zero estimates of Masser, Philippon and Wüstholz and the famous criterion for algebraic independence of Philippon. It raises fascinating new questions. One particular case that I would like to solve is showing that there are at least two prime numbers whose logarithms are algebraically independent, meaning that their logarithms are not linked by a polynomial relation with integer coefficients. Progress on this topic would probably generalize to more numbers and later extend to commutative algebraic groups towards a more general conjecture of Grothendieck. A solution to Schanuel's conjecture would have impact on all of number theory as it would for example solve a famous conjecture of Iwasawa theory called Leopoldt's conjecture. The second problem has older roots. The question of approximating a real number, like pi, by rational numbers has been central since the birth of mathematics. Nowadays, we are interested for example in simultaneous rational approximation to families of real numbers. There are general results which tell us how good an approximation we can get and, for almost all families of real numbers, one cannot do much better. However, there are exceptions and it is important to understand them. Recently, W.M. Schmidt and L. Summerer developed a general theory that should encompass all results on this topic. My first goal is to show that their description is essentially the best possible. A surprise is that unexpectedly good approximations may also exist for points in geometric progressions or more generally for points belonging to an algebraic curve (meaning that they essentially depend of just one parameter). I showed this in the case of points of a non-degenerate conic, but the curves of higher degree are a complete mystery for the moment. The study of rational approximation to points in geometric progression is particularly important because it is related, through a method of Davenport and Schmidt, to approximations to real numbers by algebraic integers of a given degree. A better understanding of this situation could eventually shed light on an important question of Wirsing. This problem is also linked with the first as it concerns understanding when certain quantities can be simultaneously small.

我的研究计划解决了数量理论的两个基本问题。 第一个涉及通常指数函数值的代数独立性。这里的Graal是Schanuel的一般合同，其中包含所有已知结果，并且所有公认的合同在这些价值观上。例如，这意味着Lindemann在1882年证明了数字PI的超越性，这反过来又表明了古希腊圆圈的不可能。在2001年，我提供了沃尔德斯米特（M. Waldschmidt）造成的辅助功能的构建应足以攻击该合同。这似乎是解决协议的第一个现实计划。它将问题降低到我所说的“小价值估计”，当多项式可以在高度结构化集合的许多点上采用小值时，分析的问题。它呼吁结果将涵盖Masser，Philippon和Wüstholz的精彩零估计以及菲利利代数独立的著名标准。它提出了令人着迷的新问题。我想解决的一种特殊情况是表明，至少有两个质数，它们的对数在代数上是独立的，这意味着它们的对数与多项式关系与整数系数没有联系。该主题的进展可能会概括为更多的数字，后来又扩展到了更一般的Grothendieck猜想。解决雪泽尔的构想的一种解决方案将对所有数字理论产生影响，例如，它可以解决一个著名的Iwasawa理论概念，称为Leopoldt的概念。 第二个问题具有较旧的根源。自数学诞生以来，按照理性数字近似实际数字（例如PI）的问题一直是核心。如今，我们感兴趣的是，与真实数字家庭的简单合理近似。有一些总体结果告诉我们，我们能获得多么近似，几乎对于所有实数家庭，一个人都做得更好。但是，有例外，重要的是要了解它们。最近，W.M.施密特（Schmidt）和L. Summerer（L. Summerer）开发了一种一般理论，该理论应涵盖该主题的所有结果。我的第一个目标是证明他们的描述本质上是最好的。令人惊讶的是，在几何进程中或更一般地，对于属于代数曲线的点，也可能存在良好的近似值（这本质上仅仅取决于一个参数）。我在非分类圆锥的点的情况下展示了这一点，但是较高程度的曲线目前是一个完全的谜。对几何进程中点的理性近似的研究尤其重要，因为它通过Davenport和Schmidt的方法与给定程度的代数整数与实数的近似相关。对这种情况的更好理解最终可能会阐明一个重要的杂物问题。该问题也与第一个问题有关，因为它涉及了解某些数量何时很小的理解。