指数超越幂的Schanuel性质

Schanuel猜想与z和w超越的代数幂z^w和w^z

Schanuel's conjecture and algebraic powers z^w and w^z with z and w transcendental

我们简要介绍了超越数理论的发展历史，包括Schanuel猜想（S）。假设（S），我们证明如果z和w是复数，不是0或1，且z^w和w^z是代数的，则z和w要么都是有理数，要么都是超越数。一个推论是，如果（S）为真，那么我们可以找到四个不同的超越正实数x，y，s，t，使得三个数x^y=/=y^x和s^t=t^s都是整数。另一个应用（可能是已知的）是（S）意味着数sqrt（2）^sqrt（2）^sqrt（2），i^i^i和i^e^pi的超越。我们还证明了如果（S）成立且a^a^z=z，其中a=/=0是代数的且z是无理数，则z是超越的.

Ax-Schanuel Type Theorems on Functional Transcendence via Nevanlinna Theory

We will apply Nevanlinna Theory to prove several Ax-Schanuel type Theorems for functional transcendence when the exponential map is replaced by other meromorphic functions. We also show that analytic dependence will imply algebraic dependence for certain classes of entire functions. Finally, some links to transcendental number theory and geometric Ax-Schanuel Theorem will be discussed.

Enlarged mixed Shimura varieties, bi-algebraic system and some Ax type transcendental results

We develop a theory of enlarged mixed Shimura varieties, putting the universal vectorial bi-extension defined by Coleman into this framework to study some functional transcendental results of Ax type. We study their bi-algebraic systems, formulate the Ax-Schanuel conjecture and explain its relation with the Ax logarithmique theorem and the Ax-Lindemann theorem which we shall prove. We also prove the whole Ax-Schanuel conjecture for the unipotent part. All these bi-algebraic and transcendental results generalize their counterparts for mixed Shimura varieties. In the end we briefly discuss about…

位置记数法中最左边数字的联合分布与Schanuel猜想

Joint distribution of leftmost digits in positional notation and Schanuels's conjecture

本文研究了位置记数法中最左边数字的联合分布，并证明了在特定条件下，不同基数的对数是无理无关的，与Schanuel猜想相关。

Effective transcendental Zilber-Pink for variations of Hodge structures

We prove function field versions of the Zilber-Pink conjectures for varieties supporting a variation of Hodge structures. A form of these results for Shimura varieties in the context of unlikely intersections is the following. Let $S$ be a connected pure Shimura variety with a fixed quasiprojective embedding. We show that there is an explicitly computable function $B$ of two natural number arguments so that for any field extension $K$ of the complex numbers and Hodge generic irreducible proper subvariety $X \subsetneq S_K$, the set of nonconstant points in the intersection of $X$ with the unio…

一个特殊的连分数

An Unusual Continued Fraction

我们考虑实数$σ$的连分数展开式$[a_0, a_1, a_2,\ldots] = [1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,\ldots]$，其中$a_i$是$2$除以$i+1$的最大幂.计算了$σ^2$的无理性测度，证明了$σ^2$（和$σ$）都是超越数.我们还表明，某些部分的$σ^2$的双指数增长，从而证实了一个猜想的汉娜和威尔逊。