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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=611431
• 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论中的两个基本问题。 第一个是关于通常的指数函数值的代数无关性。这里的Graal是Schanuel的一个非常普遍的猜想，它包含了关于这些值的所有已知结果和所有普遍接受的猜想。例如，它暗示着数字pi的超越性，林德曼在1882年证明了这一点，这反过来又表明了古希腊问题将圆平方的可能性。2001年，我证明了由M.Waldschmidt构造的辅助函数应该足以反驳这一猜想。这似乎是解决这一猜想的第一个现实方案。它将问题简化为我所说的“小值估计”，即分析一个多项式何时可以在高度结构化的集合的许多点上取小值的问题。它要求得到一个结果，它包含了Masser，Philippon和Wüstholz的奇妙的零估计，以及Philippon代数独立性的著名判据。它提出了一些引人入胜的新问题。我想要解决的一个特殊情况是，至少有两个素数的对数是代数独立的，这意味着它们的对数不是由具有整数系数的多项式关系联系在一起的。这一主题的进展可能会推广到更多的数，然后扩展到交换代数群，从而形成更一般的Grothendieck猜想。Schanuel猜想的解决方案将对整个数论产生影响，例如，它将解决岩泽理论的一个著名猜想--Leopoldt猜想。 第二个问题有着更古老的根源。自数学诞生以来，用有理数来逼近像pi这样的实数的问题就一直是核心问题。如今，我们对实数族的同时有理逼近感兴趣。有一些普遍的结果告诉我们，我们可以得到多么好的近似值，并且，对于几乎所有的实数族，没有人能做得更好了。然而，也有例外，理解它们是很重要的。最近，W.M.Schmidt和L.Summerer发展了一个普遍的理论，应该涵盖这个主题的所有结果。我的第一个目标是表明他们的描述基本上是最好的。令人惊讶的是，对于几何级数中的点，或者更一般地，对于属于代数曲线的点，也可能存在意想不到的良好逼近(这意味着它们基本上只依赖于一个参数)。我在非退化二次曲线的点的情况下说明了这一点，但目前高次曲线完全是一个谜。几何级数中点的有理逼近的研究特别重要，因为它通过Davenport和Schmidt的方法与给定次数的代数整数对实数的逼近有关。对这种情况的更好理解最终可能会揭示Wirsing的一个重要问题。这个问题也与第一个问题联系在一起，因为它涉及到理解何时某些量可以同时很小。