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论中的两个基本问题。 第一个处理代数独立的值的通常指数函数。Graal在这里是一个非常普遍的猜想Schanuel其中包含所有已知的结果和所有普遍接受的aptutures对这些价值观。例如，它意味着超越数π，证明了林德曼在1882年，这反过来又表明了不可能的古希腊问题的平方圆。在2001年，我证明了由M.瓦尔德舒格尔应该足以反驳这一猜想。这似乎是解决猜想的第一个现实的方案。它将问题简化为我称之为“小值估计”的问题，即分析多项式何时可以在高度结构化的集合的许多点处取小值的问题。它要求一个结果，将包括精彩的零估计的Masser，Philippon和Wüstholz和著名的标准代数独立的Philippon。它提出了一些有趣的新问题。我想解决的一个特殊情况是证明至少有两个素数的代数独立，这意味着它们的代数不通过整数系数的多项式关系联系起来。在这一主题上的进展可能会推广到更多的数字，后来扩展到交换代数群对一个更一般的猜想格罗滕迪克。 解决Schanuel的猜想将影响所有的数论，因为它将例如解决一个著名的猜想岩泽理论称为利奥波德猜想。 第二个问题有着更古老的根源。自数学诞生以来，用有理数逼近一个真实的数（如π）的问题一直是核心问题。 如今，我们对真实的数字族的同时有理逼近感兴趣。有一般的结果告诉我们如何好的近似，我们可以得到，几乎所有家庭的真实的号码，不能做得更好。 但也有例外，理解它们很重要。 最近，W.M.施密特和L. Summerer开发了一个一般性的理论，应该包括所有的结果在这个问题上。我的第一个目标是证明他们的描述基本上是最好的。令人惊讶的是，几何级数中的点或更一般地属于代数曲线的点也可能存在意想不到的好近似（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证明了这一点，但是高次曲线目前完全是个谜。研究有理逼近点的几何级数是特别重要的，因为它是相关的，通过达文波特和施密特的方法，近似真实的数字的代数整数的一个给定的程度。更好地了解这种情况最终可能揭示一个重要的问题Wirsing。 这个问题也与第一个问题有关，因为它涉及理解某些量何时可以同时很小。