Q/Z的群环及一个除数问题的应用

Zsigmondy's theorem and primitive divisors of the Lucas and Lehmer sequences in polynomial rings

In this paper, we obtain analogues of Zsigmondy's theorem and the primitive divisor results for the Lucas and Lehmer sequences in polynomial rings of several variables.

亏格 1 曲线上 $\mathbb{Q}$-除子的截面环

Section Rings of $\mathbb{Q}$-Divisors on Genus $1$ Curves

本文研究椭圆曲线上有理除子的截面环，推广已有亏格零及堆叠曲线相关结果，给出两点支集有效除子的生成元与关系首项，并界定任意有限点支集情形的生成元次数。

整数环中除数计数公式：除数函数d₀(n)的推广

A formula of counting divisors in integers rings: a generalization of the divisor function $d_0(n)$

本文通过建立整数环中理想结构与加性组合学理论之间的联系，证明了在具有循环类群的整数环中，每个非零理想均可分解为一个最大主因子和一个总长度受达文波特常数限制的理想乘积。通过这种分解，找到了理想I=(α, β)生成元的除数。本文的核心成果是利用有限阿贝尔群上的特征理论，推导出一个闭合公式，用于精确计数给定序列的零和子序列数目。该公式在建立主理想与零和序列之间的对应关系下，提供了对任意给定理想主理想除数的精确计数，从而计数理想I=(α, β)生成元的公共除数。这一结果构成了从唯一分解环到具有有限类群的任何迪克因环的除数函数d₀(n)的自然推广。最后，本文基于这些零和子序列的计数，对O_K中的不可约元进行了特征化。

Section rings of $\mathbb{Q}$-divisors on minimal rational surfaces

We give bounds on the degree of generation and relations of section rings associated to arbitrary $\mathbb{Q}$-divisors on projective spaces of all dimensions and Hirzebruch surfaces. For section rings of effective $\mathbb{Q}$-divisors on projective spaces, we find the best possible bound on the degrees of generators and relations.

On the mean value of the functions related to the divisor function on the ring of polynomials over a finite field

Let $ \mathbb{F}_q[T]$\, be the ring of polynomials over a finite field $ \mathbb{F}_q $. Let $ g: \mathbb{F}_q[T] \rightarrow \mathbb{R} $ be a multiplicative function such that for any irreducible polynomial $ P $ over $ \mathbb{F}_q $ and any $ k \ge 1 $, the equality $ d_k = g (P ^ k) $ holds for some arbitrary sequence of reals $\{d_k\}_{k=1}^{\infty}$. In this paper, we get an explicit formula for the sum $$ T (N) = \sum\limits_{\substack{°F=N F \text{ is monic}}}{g (F)}, $$ and also derive different asymptotics when this sum in cases of $ q \to \infty; \ q \to \infty, \ N \to \infty; \ …

A Survey on Fixed Divisors

In this article, we compile the work done by various mathematicians on the topic of the fixed divisor of a polynomial. This article explains most of the results concisely and is intended to be an exhaustive survey. We present the results on fixed divisors in various algebraic settings as well as the applications of fixed divisors to various algebraic and number theoretic problems. The work is presented in an orderly fashion so as to start from the simplest case of $\Z,$ progressively leading up to the case of Dedekind domains. We also ask a few open questions according to their context, which …