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。 第一个涉及同时合理逼近家庭的真实的号码。 一般的结果确保在一个精确的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。 这就是发生的情况，例如家庭的代数数，由于子空间定理的W. M。施密特，这一事实具有重要的后果，解决丢番图方程。 这也是有理数的指数由于A的结果而发生的情况。贝克 我研究的一个目标是在一个adelic框架下将这个结果推广到代数数的指数。 另一方面，也有一些数族，它们的近似值比预测的要好得多。 本文介绍了W. M.施密特和L. Summerer，是描述关于有理逼近的所有可能的行为。 对于经典问题，这归结为研究依赖于一个参数的特殊凸体族的连续极小值。 在这种情况下的理论是完全令人满意的，并导致显着的进展。 为了更进一步，将期望处理取决于若干参数的凸体族。 我认为，理论应该很好地扩展在框架提供的功能领域与一个无限领域的常数。 然而，在有理数上，这带来了具有挑战性的问题。 为了解决这些问题，我研究了一类极小值具有令人惊讶的代数性质的例子。这项研究与利特尔伍德的一个著名猜想有关，可能会给它带来新的启发。我的程序的第二部分涉及代数的独立性的值通常的指数函数。这里的最终目标包括一个一般性的猜想Schanuel其中包含所有已知的结果，如超越的数字pi，以及所有普遍接受的aptutures对这些价值观。2001年，我证明了M.瓦尔德舒格尔应该足以反驳这一猜想。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。到目前为止，我在这方面只取得了部分成果，由我的博士生L。Ghidelli和V. Nguyen。 为了更进一步，更好地理解这个辅助函数将是有用的，如果可能的话，能够显式地计算给定次数的辅助函数。 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，因为它可以提供绕过我们方法的实际局限性的钥匙。