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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759212
• 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互关联的部分。第一个涉及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，无法做得更好。例如，由于 W.M. 的子空间定理，代数数族就会发生这种情况。施密特，这一事实对于求解丢番图方程具有重要影响。由于 A. Baker 的结果，这也是有理数指数发生的情况。我的研究目标之一是在 adelic 框架中将这一结果扩展到代数数的指数。另一方面，有些数字的近似值比预测的要好得多。 W.M. 最近提出的参数化数字几何的目标Schmidt 和 L. Summerer 的观点是关于有理近似来描述所有可能的行为。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最小值。这个案例的理论是完全令人满意的，并取得了显着的进展。更进一步，需要根据几个参数来处理凸体族。我认为该理论应该在具有无限常数域的函数域提供的框架中很好地扩展。然而，在有理数之上，这带来了具有挑战性的问题。为了解决这些问题，我研究了一类示例，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，可以为它带来新的线索。 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沙努埃尔的一般猜想，其中包含所有已知的结果，例如数字 pi 的超越，以及对这些值的所有普遍接受的猜想。 2001 年，我证明了 M. Waldschmidt 提出的辅助函数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处取小值的问题。到目前为止，我只取得了实现这一目标的部分成果，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，并且如果可能的话，能够针对给定的度数显式地计算它将会很有用。不久前，我注意到这个函数有一些意想不到的消失特性。我建议深化这一分析，因为它可以提供规避我们方法的实际局限性的关键。