The cyclotomic polynomial topologically

Some $q$-supercongruences modulo the fourth power of a cyclotomic polynomial

In terms of the creative microscoping method recently introduced by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials, we establish a $q$-supercongruence with two parameters modulo $[n]Φ_n(q)^3$. Here $[n]=(1-q^n)/(1-q)$ and $Φ_n(q)$ is the $n$-th cyclotomic polynomial in $q$. In particular, we confirm a recent conjecture of Guo and give a complete $q$-analogue of Long's supercongruence. The latter is also a generalization of a recent $q$-supercongruence obtained by Guo and Schlosser.

$q$-Supercongruences modulo the fourth power of a cyclotomic polynomial via creative microscoping

By applying Chinese remainder theorem for coprime polynomials and the "creative microscoping" method recently introduced by the author and Zudilin, we establish parametric generalizations of three $q$-supercongruences modulo the fourth power of a cyclotomic polynomial. The original $q$-supercongruences then follow from these parametric generalizations by taking the limits as the parameter tends to $1$ (l'Hôpital's rule is utilized here). In particular, we prove a complete $q$-analogue of the (J.2) supercongruence of Van Hamme and a complete $q$-analogue of a "divergent" Ramanujan-type supercon…

多项式的分圆计数

Cyclotomic enumeration of polynomials

本文利用分圆恒等式研究有限域上首一多项式的计数，给出 gcd 无 r 次幂因子的 d 元组数量，以及不可约因子重数属于两互素整数生成幺半群的多项式数量。

二元分圆多项式：通过字表示与算法

Binary Cyclotomic Polynomials: Representation via Words and Algorithms

本文研究二元分圆多项式，将其系数向量视为三元字母表上的字，设计了一种线性时间的最优算法以计算其紧凑表示，并推广到二维数值半群相关多项式。

多维级数模分圆多项式六次幂的 $q$-超同余式

$q$-Supercongruences for multidimensional series modulo the sixth power of a cyclotomic polynomial

本文结合 El Bachraoui 引理、创造性微缩法及互素多项式中国剩余定理，证明模分圆多项式六次幂的二重与三重级数 $q$-超同余式，并导出对应 Van Hamme (D.2) 的超同余式。

循环欧拉-马霍尼多项式

Cyclotomic Euler-Mahonian polynomials

本文推导了循环欧拉-马霍尼多项式的公式，并获得了Wachs公式的I-类比及奇数情况的结果。

Coefficients of a relative of cyclotomic polynomials

Let $N=p_1p_2... p_n$ be a product of $n$ distinct primes. Define $P_N(x)$ to be the polynomial $(1-x^N)\prod_{1\leq i<j\leq n}(1-x^{N/(p_ip_j)})/\prod_{i=1}^n (1-x^{N/p_i})$. (When $n=2$, $P_{pq}(x)$ is the $pq$-th cyclotomic polynomial, and when $n=3$, $P_{pqr}(x)$ is $(1-x)$ times the $pqr$-th cyclotomic polynomial.) Let the height of a polynomial be the maximum absolute value of one of its coefficients. It is well known that the height of $Φ_{pq}(x)$ is 1, and Gallot and Moree showed that the same is true for $P_{pqr}(x)$ when $n=3$. We show that the coefficients of $P_N(x)$ depend mainly …

Cyclotomic factors of the descent set polynomial

We introduce the notion of the descent set polynomial as an alternative way of encoding the sizes of descent classes of permutations. Descent set polynomials exhibit interesting factorization patterns. We explore the question of when particular cyclotomic factors divide these polynomials. As an instance we deduce that the proportion of odd entries in the descent set statistics in the symmetric group S_n only depends on the number on 1's in the binary expansion of n. We observe similar properties for the signed descent set statistics.

关于利用分圆映射构造 bent 函数

On constructing bent functions from cyclotomic mappings

提出基于分圆映射并结合 Dillon、Niho、Kasami 分支函数的 bent 函数通用构造，得到新无限族及其对偶，并证明其 EA 不等价性与对既有构造的推广。

具有许多 cyclotomic 因子的掩码多项式下界

Lower bounds for mask polynomials with many cyclotomic divisors

研究具有多个 cyclotomic 因子的掩码多项式最小规模问题，发现当 k≥2 时，最优配置不总是具有简单的纤维结构，结果对 Cantor 集 Favard 长度、Fuglede 谱集猜想和整数平铺猜想有应用价值。

A classification of all 1-Salem graphs

One way to study certain classes of polynomials is by considering examples that are attached to combinatorial objects. Any graph $G$ has an associated reciprocal polynomial $R_G$, and with two particular classes of reciprocal polynomials in mind one can ask the questions: (a) when is $R_G$ a product of cyclotomic polynomials (giving the cyclotomic graphs)? (b) when does $R_G$ have the minimal polynomial of a Salem number as its only non-cyclotomic factor (the non-trival Salem graphs)? Cyclotomic graphs were classified by Smith in 1970. Salem graphs are `spectrally close' to being cyclotomic, i…

非负循环分圆倍式的尖锐直径界

Sharp Diameter Bounds for Nonnegative Cyclotomic Multiples

本文证明非负循环分圆倍式的支撑直径下界为(p-1)N/p，并刻画等式情形，证实Steinberger猜想，进而建立Coven-Meyerowitz直径界及其改进。

关于分圆标准分次交换代数的 h-多项式

On the $h$-polynomials of cyclotomic standard graded commutative algebras

本文研究分圆标准分次交换代数的 h-多项式何时为 CI 型，证明了当 h_R(1) 为 1、4、6 或素数时必为 CI 型，并构造了 h_R(1) 为不小于 8 的非素数时的非 CI 型反例。

On Symmetric But Not Cyclotomic Numerical Semigroups

A numerical semigroup is called cyclotomic if its corresponding numerical semigroup polynomial $P_S(x)=(1-x)\sum_{s\in S}x^s$ is expressable as the product of cyclotomic polynomials. Ciolan, García-Sánchez, and Moree conjectured that for every embedding dimension at least $4$, there exists some numerical semigroup which is symmetric but not cyclotomic. We affirm this conjecture by giving an infinite class of numerical semigroup families $S_{n, t}$, which for every fixed $t$ is symmetric but not cyclotomic when $n\ge \max(8(t+1)^3,40(t+2))$ and then verify through a finite case check that the n…

分圆猜想的证明与几乎摩尔有向图的不存在性

A proof of the cyclotomic conjecture and the non-existence of almost Moore digraphs

本文证明了Gimbert提出的分圆猜想，由此推出有向度直径问题中最大出度d>1、直径k>2的几乎摩尔有向图不存在。

Chinese Remainder Theorem for Cyclotomic Polynomials in $\mathbf{Z}[X]$

By the Chinese remainder theorem, the canonical map \[Ψ_n: R[X]/(X^n-1)\to \oplus_{d|n} R[X]/Φ_d(X)\] is an isomorphism when $R$ is a field whose characteristic does not divide $n$ and $Φ_d$ is the $d$th cyclotomic polynomial. When $R$ is the ring $\mathbf{Z}$ of rational integers, this map is injective but not surjective. In this paper, we give an explicit formula for the elementary divisors of the cokernel of $Ψ_n$(when $R=\mathbb{Z}$) using the prime factorisation of $n$. We also give a pictorial algorithm using Young Tableaux that takes $O(n^{3+ε})$ bit operations for any $ε> 0$ to determi…

Cyclotomic and simplicial matroids

Two naturally occurring matroids representable over Q are shown to be dual: the {\it cyclotomic matroid} $μ_n$ represented by the $n^{th}$ roots of unity $1,ζ,ζ^2,...,ζ^{n-1}$ inside the cyclotomic extension $Q(ζ)$, and a direct sum of copies of a certain simplicial matroid, considered originally by Bolker in the context of transportation polytopes. A result of Adin leads to an upper bound for the number of $Q$-bases for $Q(ζ)$ among the $n^{th}$ roots of unity, which is tight if and only if $n$ has at most two odd prime factors. In addition, we study the Tutte polynomial of $μ_n$ in the case …

Cyclotomic expansions for $\mathfrak{gl}_N$ knot invariants via interpolation Macdonald polynomials

In this paper we construct a new basis for the cyclotomic completion of the center of the quantum $\mathfrak{gl}_N$ in terms of the interpolation Macdonald polynomials. Then we use a result of Okounkov to provide a dual basis with respect to the quantum Killing form (or Hopf pairing). The main applications are: 1) cyclotomic expansions for the $\mathfrak{gl}_N$ Reshetikhin--Turaev link invariants and the universal $\mathfrak{gl}_N$ knot invariant; 2) an explicit construction of the unified $\mathfrak{gl}_N$ invariants for integral homology 3-spheres using universal Kirby colors. These results …

利用分圆多项式重探骰子重标记问题

Revisiting Dice Relabeling using Cyclotomic Polynomials

本文围绕 Gallian–Rusin 骰子重标记问题，利用分圆多项式确定 n=2 且 m 为三素数乘积时的重标记计数，并推广到异面骰子分解及素数幂情形。

模格安全性（第四部分）：2-幂分圆域上Module-LWE的概率多项式量子攻击

Module Lattice Security (Part IV): Probabilistic Polynomial Quantum Attack on Module-LWE over 2-Power Cyclotomics

本文提出针对ML-KEM及相关2-幂分圆格方案的量子攻击，结合前三部分给出算法并验证ML-KEM-1024的近似因子满足γ≤21<q/2=1664.5，成功概率≥0.99；通过主理想问题的塔式分解实现多项式时间量子算法，并扩展到Falcon、Hawk和NTRU。

分圆生成函数第四矩定理的量化

Quantification of the Fourth Moment Theorem for Cyclotomic Generating Functions

本文针对概率生成函数根为实数或位于单位圆上的随机变量序列，利用Stein-Tikhomirov方法统一建立了定量正态逼近界，其中分圆情形需额外引入第四累积量或矩。

传递算子与强幂环形图的独立多项式

Transfer Operators and Independence Polynomials for Strong Powers of Circulant Graphs

本文研究环形图强幂中的独立集，通过传递矩阵方法，分离内层和外层约束，得到等变于二面体群作用的传递算子，并证明其特征多项式分解为异常和环状部分，通过C7验证结果。