On the Area of the Fundamental Region of a Binary Form Associated with Algebraic Trigonometric Quantities

On the Automorphism Group of a Binary Form Associated with Algebraic Trigonometric Quantities

Let $F(x, y)$ be a binary form of degree at least three and non-zero discriminant. In this article we compute the automorphism group $\operatorname{Aut} F$ for four families of binary forms. The first two families that we are interested in are homogenizations of minimal polynomials of $2\cos\left(\frac{2π}{n}\right)$ and $2\sin\left(\frac{2π}{n}\right)$, which we denote by $Ψ_n(x, y)$ and $Π_n(x, y)$, respectively. The remaining two forms that we consider are homogenizations of Chebyshev polynomials of first and second kinds, denoted $T_n(x, y)$ and $U_n(x, y)$, respectively.

On generalized trigonometric functions and series of rational functions

Here we introduce a way to construct generalized trigonometric functions associated with any complex polynomials, and the well known trigonometric functions can be seen to associate with polynomial $x^2-1$. We will show that those generalized trigonometric functions have algebraic identities which generalizes the well known $\sin^2(x)+\cos^2(x)=1$. One application of the generalized trigonometric functions is evaluating infinite series of rational functions.

拉马努金和的代数推广

An Algebraic Generalization of the Ramanujan Sum

本文通过多项式取余法推广拉马努金和，将其表示为带互质条件的有限三角和，还给出了带奇偶条件的Levenshtein码大小的显式公式，解决了SVT码的重要开放问题。

关于代数无关性的一个判据及其变体

On a criterion for algebraic independence and its variants

本文系统研究并阐释了Elsner等人提出的代数无关性转移判据及其变体，该判据借助2n元多项式组将n个量的代数无关性转移到另外n个量。

稀疏求和的累积增长算术敏感性：超越数与代数比极限

Arithmetic sensitivity of cumulant growth in lacunary sums: transcendental versus algebraic ratio limits

本文研究稀疏三角求和的累积增长，发现其渐进行为对序列的算术结构敏感，超越数比极限情况下累积量线性增长，而代数比极限时可能呈现不同增长模式。