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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=738405
• 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连接的零件。第一个涉及对实数家族的同时理性近似。一般结果确保在精确范围内存在良好的近似值，对于大多数家庭而言，一个人不能做得更好。这是对代数数字的家庭的发生，这要归功于W.M.的子空间定理。施密特（Schmidt），这一事实对解决二芬太汀方程有重要影响。由于A. Baker的结果，这也是理性数字指数的情况。我的研究的目标之一是将此结果扩展到代数数字的指数，在Adelic框架中。另一方面，有一些数字的家庭承认近似值要比预测的要好得多。 W.M.最近引入的数字参数几何的目标Schmidt和L. Summerer将描述有关有理近似的所有可能行为。对于经典问题，这归结为研究凸体特殊家族的连续最小值，具体取决于一个参数。在这种情况下，该理论完全令人满意，并导致了显着的进步。再进一步，希望根据几个参数来治疗凸形的家族。我认为该理论应该在具有无限常数字段的功能字段提供的框架中很好地扩展。但是，在理性数字上，这带来了具有挑战性的问题。为了解决它们，我研究了一类示例，在这些示例中，最小值显示出令人惊讶的代数属性。这项研究与著名的利特伍德猜想有关，可以为此带来新的启示。 我程序的第二部分涉及通常指数函数值的代数独立性。这里的最终目标是schanuel的一般猜想，其中包含所有已知结果，例如数字PI的超越性，以及对这些值的所有公认的猜想。在2001年，我证明了沃尔德斯米特（M. Waldschmidt）造成的辅助功能的构建应足以攻击这一猜想。它将问题减少到我所说的“小价值估计”，分析何时多项式可以在高度结构化集合的许多点上采用小值的问题。到目前为止，我只为此目标获得了部分结果，该目标是由我的博士生L. Ghidelli和V. Nguyen完成的。再进一步，更好地理解这种辅助功能将很有用，并在可能的情况下能够在给定的学位上明确计算它是有用的。不久前，我注意到此功能具有一些意外的消失属性。我建议加深这种分析，因为它可以提供衡量我们方法实际局限性的关键。