Algebraic independence and Diophantine approximation

代数独立性和丢番图近似

• 批准号: RGPIN-2014-05086
• 负责人: Roy, Damien
• 金额: $ 1.68万
• 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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=611431
• 关键词: Algebraic; independence; Diophantine; approximation

My research programme addresses two fundamental problems in number theory. The first one deals with algebraic independence of values of the usual exponential function. The Graal here is a very general conjecture of Schanuel which contains all known results and all generally accepted conjectures on these values. For example it implies the transcendence of the number pi, proved by Lindemann in 1882, which in turn shows the impossibility of the ancient Greek problem of squaring the circle. In 2001, I proved that a construction of auxiliary function due to M. Waldschmidt should suffice to attack this conjecture. This seems to be the first realistic programme towards solving the 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. It calls for a result that would encompass the wonderful zero estimates of Masser, Philippon and Wüstholz and the famous criterion for algebraic independence of Philippon. It raises fascinating new questions. One particular case that I would like to solve is showing that there are at least two prime numbers whose logarithms are algebraically independent, meaning that their logarithms are not linked by a polynomial relation with integer coefficients. Progress on this topic would probably generalize to more numbers and later extend to commutative algebraic groups towards a more general conjecture of Grothendieck. A solution to Schanuel's conjecture would have impact on all of number theory as it would for example solve a famous conjecture of Iwasawa theory called Leopoldt's conjecture. The second problem has older roots. The question of approximating a real number, like pi, by rational numbers has been central since the birth of mathematics. Nowadays, we are interested for example in simultaneous rational approximation to families of real numbers. There are general results which tell us how good an approximation we can get and, for almost all families of real numbers, one cannot do much better. However, there are exceptions and it is important to understand them. Recently, W.M. Schmidt and L. Summerer developed a general theory that should encompass all results on this topic. My first goal is to show that their description is essentially the best possible. A surprise is that unexpectedly good approximations may also exist for points in geometric progressions or more generally for points belonging to an algebraic curve (meaning that they essentially depend of just one parameter). I showed this in the case of points of a non-degenerate conic, but the curves of higher degree are a complete mystery for the moment. The study of rational approximation to points in geometric progression is particularly important because it is related, through a method of Davenport and Schmidt, to approximations to real numbers by algebraic integers of a given degree. A better understanding of this situation could eventually shed light on an important question of Wirsing. This problem is also linked with the first as it concerns understanding when certain quantities can be simultaneously small.

我的研究计划涉及数论中的两个基本问题。