Diophantine approximation and transcendental number theory

丢番图近似和超越数论

• 批准号: RGPIN-2019-05618
• 负责人: Roy, Damien
• 金额: $ 1.53万
• 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: 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=677855
• 关键词: Diophantine; approximation; transcendental; number; theory

My research programme is divided in two connected parts. The first one deals with simultaneous rational approximation to families of real numbers. General results ensure the existence of good approximations within a precise range and, for most families, one cannot do better. This is what happens for example for families of algebraic numbers, thanks to the subspace theorem of W.M. Schmidt, and this fact has important consequences for solving Diophantine equations. This is also what happens for exponentials of rational numbers thanks to a result of A. Baker. One goal of my research is to extend this result to exponentials of algebraic numbers, in an adelic framework. On the other hand, there are families of numbers which admit much better approximations then predicted. The goal of Parametric Geometry of Numbers, recently introduced by W.M. Schmidt and L. Summerer, is to describe all possible behaviours with respect to rational approximation. For the classical problems, this boils down to studying the successive minima of special families of convex bodies depending on one parameter. The theory in this case is fully satisfactory and has led to remarkable progresses. To go further, it would be desirable to treat families of convex bodies depending on several parameters. I think that the theory should extend nicely in the framework provided by functions fields with an infinite field of constants. However, over the rational numbers, this brings challenging problems. To resolve them, I work on a class of examples where the minima show surprising algebraic properties. This research is connected to a famous conjecture of Littlewood and could bring new light on it.*******The second part of my programme deals with algebraic independence of values of the usual exponential function. The ultimate goal here consists in a general conjecture of Schanuel which contains all known results, like the transcendence of the number pi, and all generally accepted conjectures on these values. In 2001, I proved that a construction of auxiliary function due to M. Waldschmidt should suffice to attack this 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. Up to now, I obtained only partial results towards this goal, completed by work of my PhD students L. Ghidelli and V. Nguyen. To go further, it would be useful to better understand this auxiliary function and, if possible, to be able to compute it explicitly for a given degree. Sometime ago, I noticed that this function has some unexpected vanishing properties. I propose to deepen this analysis, as it could provide the key to circumvent the actual limitations of our methods.**

我的研究计划分为两个相连的部分。第一个是关于实数族的同时有理逼近。一般结果确保在精确范围内存在良好的近似值，对于大多数家庭来说，不能做得更好。这就是代数数族的情况，多亏了w·m·施密特的子空间定理，这个事实对解丢芬图方程有重要的影响。这也适用于有理数的指数这是a。Baker的结果。我研究的一个目标是将这个结果扩展到代数数的指数，在一个矩阵框架中。另一方面，也有一些数字比预测的近似得多。最近由W.M. Schmidt和L. Summerer引入的参数数几何的目标是描述关于有理逼近的所有可能的行为。对于经典问题，这可以归结为研究依赖于一个参数的特殊凸体族的连续极小值。在这种情况下，理论是完全令人满意的，并导致了显著的进展。更进一步，我们需要根据几个参数来处理凸体族。我认为这个理论可以很好地扩展到函数场提供的框架中，在无穷常数场中。然而，对于有理数，这带来了具有挑战性的问题。为了解决这些问题，我研究了一类例子，其中最小值显示出令人惊讶的代数性质。这项研究与利特尔伍德的一个著名猜想有关，可能会给这个猜想带来新的启示。*******我的程序的第二部分处理通常指数函数值的代数独立性。这里的最终目标在于Schanuel的一般猜想，它包含了所有已知的结果，比如π的超越性，以及所有关于这些值的普遍接受的猜想。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。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。到目前为止，我对这个目标只取得了部分成果，由我的博士生L. Ghidelli和V. Nguyen完成。更进一步，更好地理解这个辅助函数，如果可能的话，能够对给定的度显式地计算它，将是有用的。不久前，我注意到这个函数有一些意想不到的消失属性。我建议深化这一分析，因为它可以提供规避我们的方法的实际局限性的关键