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：来自先验数理论的引人入胜的猜想断言，代数数的Q线性独立的对数在Q上是独立的。例如，其解决方案将暗示Leopoldt对任何prime数字p的P-ADIC调节器的非构成构成。 我的主要目标是证明代数数的任何足够大的Q线性独立对数都包含两个代数独立的元素。几年前，我表明，原则上，众所周知的结构足以以一系列多项式的形式捕获所有所需的信息，从而从代数组的GA x gm的有限生成的亚组上占据了很小的值。 这需要新的结果，这些结果将涵盖菲利普的代数独立性和实际零估计标准。最近，我能够证明对一维组的GA和GM的这种形式的非平凡估计，并且我打算开发的二维组GA X GM有部分结果。