托勒密定理与西姆松线定理的等价性及折射定律的一个初等证明

托勒密定理与西姆松线定理是两个有名的经典定理,蔡聪明在[1]中对托勒密定理及其有关性质做了细致的综述,但从西姆松线定理与托勒密定理的关系入手,更容易看清这两个经典定理的实质.     

F_q[x]上的素数定理和Dirichlet定理及最小范数不可约多项式

讨论了F_q[x]上的zeta函数和L函数的解析性质,并在不假定黎曼猜想的情况下,导出了F_q[x]上的多项式环及其算术级数中不可约多项式的分布.然后,通过一系列的技术性处理,给出了算术级数中不可约多项式的最小范数的估计.成功地把素数定理及Dirichlet定理推广到了F_q[x]中,最重要的是,对应于最小素数问题,得到的最小范数的估计值本质上要比有理整数环上假定黎曼猜想情况下所推得的结果还好.     

Fq[x]上的素数定理和Dirichlet定理及最小范数不可约多项式

讨论了Fq[x]上的zeta函数和L函数的解析性质,并在不假定黎曼猜想的情况下,导出了Fq[x]上的多项式环及其算术级数中不可约多项式的分布.然后,通过一系列的技术性处理,给出了算术级数中不可约多项式的最小范数的估计.成功地把素数定理及Dirichlet定理推广到了Fq[x]中,最重要的是,对应于最小素数问题,得到的最小范数的估计值本质上要比有理整数环上假定黎曼猜想情况下所推得的结果还好.     

Cauchy-Binet定理与Laplace定理的等价证明

本文运用分块矩阵及多元多项式的性质对行列式求值中的Cauchy-Binet 定理与Laplace 定理给出了等价证明.     

费尔马小定理的一个初等证明

利用多元多项式定理,给出费尔马小定理的一种完全初等的证明.     

恒等式的几何意义及其组合证明

给出G auss系数的定义及其几何意义,对一些G auss系数恒等式给出了组合分析的证明,并且相应地给出它们的几何解释.     

Newton公式和Vieta定理的等价性

利用待定系数法和数学归纳法证明Newton公式和Vieta定理的等价性.     

Kantor定理成立的必要条件及其应用

本文绘出了 Ｋａｎｔｏｒ定理成立的必要条件,举例说明了对一般域的有限次扩张, Ｋａｎｔｏｒ定理可能不成立,并提出用Ｋａｎｔｏｒ定理判定有限域上多项式方程在该域上有解的新方法．     

Kantor定理成立的必要条件及其应用

本文给出了Kantor定理成立的必要条件,举例说明了对一般域的有限次扩张,Kantor定理可能不成立,并提出用Kantor定理判定有限域上多项式方程在该域上有解的新方法.     

唯一补格中与分配性的等价条件和关于Peirces定理的证明

本文首先提出唯一补格与原子对偶原子相关的充要条件;然后对Peirces定理第4步证明提出了新的方法;进而提出并证明了唯一补格中与分配性的四个等价条件;本文最后提出了在唯一补格中可以推出分配性的四个等价条件,并逐一进行了证明。     

Bivariate identities related to Chebyshev and Dickson polynomials over Fq

Let ${\mathbb{F}}_q$ be a field of q elements, where q is a power of an odd prime p. The polynomial $f(y)\in {\mathbb{F}}_q[y]$ defined by$f(y):={(1+\sqrt{y})}^{(q+1)/2}+{(1-\sqrt{y})}^{(q+1)/2}$ has the property that$f(1-y)=\rho (2)f(y),$ where ρ is the quadratic character on ${\mathbb{F}}_q$. This univariate identity was applied to prove a recent theorem of N. Katz. We formulate and prove a bivariate extension, and give an application to quadratic residuacity.     

Explicit factorizations of generalized Dickson polynomials of order 2m via generalized cyclotomic polynomials over finite fields

We derive explicit factorizations of generalized cyclotomic polynomials of order $2^m$ and generalized Dickson polynomials of the first kind of order $2^m$ over finite field $F_q$.     

Reducibility of translates of Dickson polynomials

Let be a field and . The Dickson polynomial is characterized by the equation . We prove that is reducible if and only if there is a prime such that for some , or and for some . This result generalizes the well-known reducibility criterion for binomials; and it provides a reducibility criterion for where denotes the Chebyshev polynomial of degree .

     

An explanation for the similar appearance of two identities involving Dickson polynomials

This article explains the similar appearance of two polynomial identities involving Dickson polynomials in char. 2, one found by Abhyankar, Cohen and Zieve, and the other found by the author.     

Necessary conditions for reversed Dickson polynomials of the second kind to be permutational

In this paper, we present several necessary conditions for the reversed Dickson polynomial $E_n(1,x)$ of the second kind to be a permutation of ${\mathbb{F}}_q$. In particular, we give explicit evaluation of the sum ${\sum }_{a\in {\mathbb{F}}_q}{E}_n(1,a)$.     

Skew Hadamard Difference Sets from Dickson Polynomials of Order 7

Skew Hadamard difference sets have been an interesting topic of study for over 70 years. For a long time, it had been conjectured the classical Paley difference sets (the set of nonzero quadratic residues in where ) were the only example in Abelian groups. In 2006, the first author and Yuan disproved this conjecture by showing that the image set of is a new skew Hadamard difference set in with m odd, where denotes the first kind of Dickson polynomials of order n and . The key observation in the proof is that is a planar function from to for m odd. Since then a few families of new skew Hadamard difference sets have been discovered. In this paper, we prove that for all , the set is a skew Hadamard difference set in , where m is odd and . The proof is more complicated and different than that of Ding‐Yuan skew Hadamard difference sets since is not planar in . Furthermore, we show that such skew Hadamard difference sets are inequivalent to all existing ones for by comparing the triple intersection numbers.