有限域上的Chebyshev作用

切比雪夫、斐波那契与卢卡斯多项式之间通过三角和建立的关系

Relations between Chebyshev, Fibonacci and Lucas polynomials via trigonometric sums

本文通过有限三角和建立斐波那契、卢卡斯与切比雪夫多项式间的新恒等式，并给出其物理解释及与图有效电阻的联系。

无需线性独立性的切比雪夫偏差

Chebyshev's bias without linear independence

在广义黎曼假设下，通过加权逆平方根的素数计数函数，证明了非平方剩余类中素数密度为1，无需对数密度或更强的零点假设。

算术级数中的截断MacMahon型q级数与切比雪夫展开式

Truncated MacMahon-type $q$-series in arithmetic progressions and Chebyshev expansions

本文研究三类有限族MacMahon型q级数，通过加权初等对称函数论证证明$B_{k,m,N,r}^{\pm}(q)$的双q二项式展开，建立$A、C$族的切比雪夫多项式展开，证明并推广Merca的相关猜想，还给出两类族$a=-2$恒等式的加权分拆证明。

利用广义Pell方程解和切比雪夫多项式求所有可表示为连续平方整数之和的平方整数

Finding all squared integers expressible as the sum of consecutive squared integers using generalized Pell equation solutions with Chebyshev polynomials

如果是$M\equiv0,9,24$或$33(mod\,72)$ ;或$M\equiv1,2$或$16(mod\,24)$;或$M\equiv11(mod\,12)$，则从$a^{2}\geq1$开始的$M$连续整数平方和的平方根$s$为整数，如果是$M\equiv3,5,6,7,8$或$10(mod\,12)$，则不能为整数。求出所有$s$整数的解需要求解以$M$为参数的变量为$a$和$s$的丢番图二次方程。如果$M$不是平方整数，则变量为$a$和$s$的丢番图二次方程被变换为广义佩尔方程，其形式取决于$M(mod\,4)$的同余值，并且其解（如果存在）对于给定的$M$值产生$a$和$s$中的所有解。根据这个广义Pell方程是否存在一个或多个基本解，在$a$和$s$中存在一个或多个解的无限分支，这些解可以简单地写成在相关简单Pell方程的基本解处求值的切比雪夫多项式的函数.如果$M$是一个平方整数，它是已知的，$M\equiv1(mod\,24)$和$M=(6n-1)^{2}$的所有整数$n$ ;然后丢番图二次方程的变量$a$和$s$减少到一个简单的差异的整数平方产生一个有限数目的解决方案在$a$和$s$的初始问题。

A structure theorem for finite fields

We present a new structure theorem for finite fields of odd order that relates multiplicative and additive structure in an interesting way. This theorem has several applications, including an improved understanding of Dickson and Chebyshev polynomials and some formulas with a number-theoretic flavor. This paper is an abridged version of two math arXiv articles by the author [arXiv:1707.06870, arXiv:1707.06877].

切比雪夫-西维斯特方法在素数理论中的综述

An expository review of the Chebyshev-Sylvester method in prime number theory

本文综述了切比雪夫和西维斯特在素数理论中发展的基本方法，通过计算实现并优化了其核心技术，为分析数论提供了教育资源。

CM 理论、极大超椭圆曲线与 Chebyshev 多项式

CM theory, maximal hyperelliptic curves, and Chebyshev polynomials

本文研究由 Chebyshev 多项式定义的超椭圆曲线，结合 Frobenius 斜率、复乘 Abel 子簇与 Jacobian 的 $2$-下降技术，给出并排除若干 $\FF_{q^2}$ 上的极大性情形。

有限域上强大斐波那契多项式

Powerful Fibonacci polynomials over finite fields

本文研究了有限域上斐波那契多项式及广义卢卡斯多项式成为完全幂或强大多项式的特征。

A large arboreal Galois representation for a cubic postcritically finite polynomial

We give a complete description of the arboreal Galois representation of a certain postcritically finite cubic polynomial over a large class of number fields and for a large class of basepoints. This is the first such example that is not conjugate to a power map, Chebyshev polynomial, or Lattès map. The associated Galois action on an infinite ternary rooted tree has Hausdorff dimension bounded strictly between that of the infinite wreath product of cyclic groups and that of the infinite wreath product of symmetric groups. We deduce a zero-density result for prime divisors in an orbit under this…

Iterated monodromy groups of rational functions and periodic points over finite fields

Let $q$ be a prime power and $ϕ$ a rational function with coefficients in a finite field $\mathbb{F}_q$. For $n \geq 1$, each element of $\mathbb{P}^1(\F_{q^n})$ is either periodic or strictly preperiodic under iteration of $ϕ$. Denote by $a_n$ the proportion of periodic elements. Little is known about how $a_n$ changes as $n$ grows, unless $ϕ$ is a power map or Chebyshev polynomial. We give the first results on this question for a wider class of rational functions: $a_n$ has lim inf $0$ when $q$ is odd and $ϕ$ is quadratic and neither Lattès nor conjugate to a one-parameter family of exceptio…