A stronger version of a question proposed by K. Mahler

Méthode de Mahler en caract{é}ristique non nulle : un analogue du Théorème de Ku. Nishioka

In 1990, Ku. Nishioka proved a fundamental theorem for Mahler's method, which is the analog of the Siegel-Shidlovskii theorem for Mahler functions. In this article, we establish a version of the theorem of Ku. Nishioka which is also valid for Mahler systems over function fields of positive characteristic. We follow the approach introduced by Denis in 1999 in a particular case. It is based on an algebraic independence criterion from Philippon. The main motivation of this work is built on the following remarkable fact discovered by Denis. Over function fields of positive characteristic, analogs …

Two-variable polynomials with dynamical Mahler measure zero

We discuss several aspects of the dynamical Mahler measure for multivariate polynomials. We prove a weak dynamical version of Boyd--Lawton formula and we characterize the polynomials with integer coefficients having dynamical Mahler measure zero both for the case of one variable (Kronecker's lemma) and for the case of two variables, under the assumption that the dynamical version of Lehmer's question is true.

度量化Mahler测度的参数化族

The parametrized family of metric Mahler measures

本文推广了Mahler测度的度量版本$M_t$，利用其给出Lehmer猜想的等价形式，证明该函数由指数函数分段构造，并提出一个关于有理数图像绘制的猜想。

n-射影空间上的Mahler公式

Mahler formula on the projective n-space

本文给出了经典Mahler公式的一种形式，即高度和测度是相对于任意维数的射影空间上的自映射的。

Mahler-Popken复杂度的二进制版本

A binary version of the Mahler-Popken complexity function

本文提出Mahler-Popken复杂度的二进制版本，定义$\\|n\\|_2$为用1和2表达n所需最少数字个数，证明其与最短加法链长度的联系，并给出显式上下界。

Bloch 循环与弱二元 Chinburg 猜想

The weak Chinburg conjecture on Mahler measures

本文通过构造有理多项式 $R_f$ 并证明其 Mahler 测度等于 $4wL'(\chi_{-f},-1)$，从而证明了弱二元 Chinburg 猜想。

论非阿基米德度量Mahler测度

On the non-Archimedean metric Mahler measure

本文提出了度量Mahler测度的非阿基米德版本$M_\infty$，证明了其取值为1当且仅当$α$为单位根，并验证了其在相应向量空间上定义了射影高度，最后给出了根式情形下的计算方法。

关于涉及导数的 Mahler 问题 B 的一个版本

On a version of the Problem B of Mahler involving derivatives

本文研究 Mahler 问题 B 的导数推广版本，证明存在不可数多个超越整函数，使其本身、反函数及导数在代数点上均取代数值。

Counting Exceptional Points for Rational Numbers Associated to the Fibonacci Sequence

If $α$ is a non-zero algebraic number, we let $m(α)$ denote the Mahler measure of the minimal polynomial of $α$ over $\mathbb Z$. A series of articles by Dubickas and Smyth, and later by the author, develop a modified version of the Mahler measure called the $t$-metric Mahler measure, denoted $m_t(α)$. For fixed $α\in \bar{\mathbb Q}$, the map $t\mapsto m_t(α)$ is continuous, and moreover, is infinitely differentiable at all but finitely many points, called {\it exceptional points} for $α$. It remains open to determine whether there is a sequence of elements $α_n\in \bar{\mathbb Q}$ such that …

阿贝尔群上的度量高度

Metric Heights on an Abelian Group

作者研究了具有三角不等式的 Mahler measure 变体 $m_t(α)$，将其通用构造推广至任意 Abelian group 上的抽象函数，证明了已知结果的类似定理，并解决了先前关于 $m_t(α)$ 的开放问题。

关于多变量超越整函数的例外集

On the Exceptional Set of Transcendental Entire Functions in Several Variables

本文证明了 $\overline{\mathbb{Q}}^m$ 的任意子集（复共轭下封闭且含原点）是 $\mathbb{C}^m$ 上有理系数超越整函数的例外集，解决了 Mahler 问题的多变量版本。

解析霍普夫代数的对偶性与阿米斯变换

Duality of analytic Hopf algebras and the Amice transform

本文构建了适用于一般巴拿赫环的解析霍普夫代数的全局版本，研究了其对偶理论，并通过基变换成现了阿米斯对偶性，不依赖于特定素数p。

Generalised Binomial coefficients and Jarden's Theorem

We prove a stronger version of Jarden's Theorem for recurrence of powers of recursive functions

Rational approximation on quadrics: a simplex lemma and its consequences

We give elementary proof of stronger versions of several recent results on intrinsic Diophantine approximation on rational quadric hypersurfaces $X\subset \mathbb{P}^n(\mathbb{R})$. The main tool is a refinement of the simplex lemma, which essentially says that rational points on $X$ which are sufficiently close to each other must lie on a totally isotropic rational subspace of $X$.