Algebraic independence and diophantine approximation

-主要主题：一个来自超越数论的有趣猜想断言，代数数的q -线性无关对数在q上是代数无关的。例如，它的解将暗示利奥波德关于任何素数p的数字域的p进调节不消失的猜想。我的主要目标是证明任何足够大的代数数的q -线性无关对数集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捕获所有所需的信息，以多项式序列的形式，在代数群Ga x Gm的有限生成子群的大集合点上取小值。这需要新的结果，既包括菲利波代数独立性准则，也包括实际的零估计。最近，我能够证明一维群Ga和Gm的这种形式的非平凡估计，并且我有二维群Ga x Gm的部分结果，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.