Diophantine approximation and transcendental number theory

我的研究计划分为两个相互关联的部分。第一章讨论实数族的同时有理逼近。一般的结果确保在一个精确的范围内存在良好的近似，对于大多数家庭来说，没有人能做得更好。例如，由于W.M.施密特的子空间定理，这就是代数数族所发生的事情，这一事实对求解丢番图方程具有重要的影响。这也是有理数的指数所发生的，这要归功于A·贝克的结果。我的研究目标之一是将这一结果推广到代数数的指数，在adelic框架下。另一方面，也有一些数族接受比预测好得多的近似值。最近由W.M.Schmidt和L.Summerer提出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，这归结为研究依赖于一个参数的特殊凸体族的逐次极小值。这种情况下的理论是完全令人满意的，并导致了显著的进步。更进一步，根据几个参数来处理凸体族是可取的。我认为，这一理论应该很好地扩展到函数域提供的框架中，函数域具有无穷大的常量域。然而，在有理数字之上，这带来了具有挑战性的问题。为了解决这些问题，我处理了一类例子，其中极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，并可能给它带来新的启发。*我的程序的第二部分处理的是通常指数函数值的代数无关性。这里的终极目标在于Schanuel的一般猜想，它包含了所有已知的结果，如pi数的超越，以及所有关于这些值的普遍接受的猜想。2001年，我证明了由M.Waldschmidt构造的辅助函数应该足以反驳这一猜想。它将问题简化为我所说的“小值估计”，即分析一个多项式何时可以在高度结构化的集合的许多点上取小值的问题。到目前为止，我在这个目标上只取得了部分成果，这是我的博士生L.Gideelli和V.Nguyen的努力完成的。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的次数。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限制的关键。

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.**