Algebraic independence and diophantine approximation

- 主要议题：超越数论中一个有趣的猜想断言，代数数的Q-线性无关代数是在Q上代数无关的。 它的解决方案将例如意味着一个猜想Leopoldt对非零的p-进调节器的一个数字领域的任何素数p.我的主要目标是要证明，任何足够大的一套Q-线性无关的代数数包含两个代数独立的元素。几年前，我证明了，原则上，众所周知的结构足以捕捉所有需要的信息，以多项式序列的形式，在代数群Ga x Gm的一个非线性生成子群的大点集上取小值。 这需要新的结果，将包括两个菲利蓬的标准代数独立性和实际零估计。最近，我能够证明非平凡的估计，这种形式的一维组Ga和Gm，我有部分结果的二维组Ga x Gm，我打算发展。

- 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.