Algebraic independence and diophantine approximation

代数独立性和丢番图近似

• 批准号: 138225-2009
• 负责人: Roy, Damien
• 金额: $ 2.04万
• 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: 2013
• 资助国家: 加拿大
• 起止时间: 2013-01-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=522381
• 关键词: Algebraic; independence; diophantine; approximation

- Main topic: A fascinating conjecture from transcendental number theory asserts that Q-linearly independent logarithms of algebraic numbers are algebraically independent over Q. Its solution would for example imply a conjecture of Leopoldt on the non-vanishing of the p-adic regulator of a number field for any prime number p. My main goal is to prove that any sufficiently large set of Q-linearly independent logarithms of algebraic numbers contains two algebraically independent elements. Some years ago, I showed that, in principle, well-known constructions are sufficient to capture all the needed information, in the form of a sequence of polynomials taking small values on large sets of points from a finitely generated subgroup of the algebraic group Ga x Gm. This requires new results that would encompass both Philippon's criterion for algebraic independence and the actual zero estimates. Recently, I was able to prove non-trivial estimates of this form for the one dimensional groups Ga and Gm, and I have partial results for the two-dimensional group Ga x Gm which I intend to develop.

- 主要主题：超越数论的一个引人入胜的猜想断言代数数的 Q 线性独立对数在代数上独立于 Q。例如，它的解决方案意味着 Leopoldt 的猜想，即对于任何素数 p，数域的 p 进数调节器不消失。 我的主要目标是证明任何足够大的 Q 线性独立代数数对数集都包含两个代数独立元素。几年前，我表明，原则上，众所周知的构造足以捕获所有需要的信息，其形式为一系列多项式，在代数群 Ga x Gm 的有限生成子群中的大点集上取小值。 这需要新的结果，其中包括菲利蓬的代数独立性准则和实际的零估计。最近，我能够证明一维群 Ga 和 Gm 的这种形式的非平凡估计，并且我对我打算开发的二维群 Ga x Gm 有了部分结果。