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: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=566942
• 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.

我的研究计划涉及数论中的两个基本问题。第一个是一般指数函数值的代数无关性。这里的Graal是Schanuel的一个非常一般的猜想，它包含了所有已知的结果和所有关于这些值的普遍接受的猜想。例如，它暗示了圆周率的超越性，这是由林德曼在1882年证明的，这反过来又表明了古希腊的圆的平方问题是不可能的。在2001年，我证明了M. Waldschmidt的辅助函数的构造应该足以攻击这个猜想。这似乎是解决这个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”，即分析多项式何时可以在高度结构化集合的许多点上取小值的问题。它需要一个包含了马泽、菲利蓬和w<s:1> stholz美妙的零估计和著名的菲利蓬代数独立性准则的结果。它提出了一些有趣的新问题。我想解决的一个特例是证明至少有两个素数它们的对数是代数无关的，也就是说它们的对数不是由整数系数的多项式关系联系起来的。这个主题的进展可能会推广到更多的数字，后来扩展到交换代数群，以实现更一般的格罗滕迪克猜想。Schanuel猜想的解会对整个数论产生影响比如它会解决岩泽理论中一个著名的猜想，利奥波德猜想。第二个问题有着更古老的根源。用有理数近似圆周率这样的实数的问题，自数学诞生以来一直是核心问题。现在，我们感兴趣的是实数族的同时有理逼近。有一些一般的结果告诉我们，我们可以得到多么好的近似，对于几乎所有的实数族，我们不能做得更好。然而，也有例外，了解它们是很重要的。最近，w·m·施密特和l·萨默尔提出了一个通用理论，它应该涵盖了关于这个主题的所有结果。我的第一个目标是展示他们的描述本质上是最好的。令人惊讶的是，对于几何级数中的点，或者更一般地说，对于属于代数曲线的点（这意味着它们本质上只依赖于一个参数），也可能存在意想不到的良好近似值。我在非退化二次曲线的点的情况下展示了这一点，但更高次的曲线目前完全是一个谜。对几何级数中点的有理逼近的研究是特别重要的，因为通过达文波特和施密特的方法，它与用给定次的代数整数逼近实数有关。对这种情况的更好理解最终可能会揭示一个重要的Wirsing问题。这个问题也与第一个问题有关，因为它涉及到理解某些数量何时可以同时很小。