Kurt Girstmair
Let $n\ge 3$ be a square-free natural number. We explicitly describe the inverses of the matrices $$ (2\sin(2πjk^*/n))_{j,k} \enspace \mbox{ and }\enspace (2\cos(2πjk^*/n))_{j,k}, $$ where $k^*$ denotes a multiplicative inverse of $k$ mod $n$ and $j,k$ run through the set $\{l; 1\le l\le n/2, (l,n)=1\}$. These results are based on the theory of Gauss sums.
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We classify all cyclotomic matrices over the Eisenstein and Gaussian integers, that is, all Hermitian matrices over the Eisenstein and Gaussian integers that have all their eigenvalues in the interval [-2, 2].
We classify all cyclotomic matrices over real quadratic integer rings and we show that this classification is the same as classifying cyclotomic matrices over the compositum all real quadratic integer rings. Moreover, we enumerate a related class of symmetric matrices; those matrices whose eigenvalues are contained inside the interval [-2,2] but whose characteristic polynomials are not in Z[x].
In this paper, by using the theory of circulant matrices we study some matrices over finite fields which involve the quadratic character and trinomial coefficients.
Cyclotomic matrices and hypergeometric functions over finite fields
本文借助有限域上的超几何函数，研究了涉及特征与二元二次型的分圆矩阵的算术性质，并证实了Zhi-Wei Sun提出的相关猜想。
We propose and justify a generalised approach to prove the polynomial reduction of the RLWE to the PLWE problem attached to the ring of integers of a monogenic number field. We prove such equivalence in the case of the maximal totally real subextension of the 4p-th cyclotomic field, with p arbitrary prime.
On new identities of Jacobi sums and related cyclotomic matrices
本文利用Jacobi和的算术性质，研究涉及Jacobi和的乘积，并揭示这些乘积与特定循环矩阵之间的联系，验证了Sun在2019年提出的猜想并获得更强的结果。
We determine all Hermitian $\mathcal{O}_{\Q(\sqrt{d})}$-matrices for which every eigenvalue is in the interval [-2,2], for each d in {-2,-7,-11,-15\}. To do so, we generalise charged signed graphs to $\mathcal{L}$-graphs for appropriate finite sets $\mathcal{L}$, and classify all $\mathcal{L}$-graphs satisfying the same eigenvalue constraints. We find that, as in the integer case, any such matrix / graph is contained in a maximal example with all eigenvalues $\pm2$.
On $p$-th cyclotomic field and cyclotomic matrices involving Jacobi sums
本文研究了p次环形矩阵的算术性质，证明了其行列式与特定多项式系数的关系。
Confidentiality and Integrity are two paramount objectives in the evaluation of information and communication technology. In this paper, we propose an arithmetic approach for designing asymmetric key cryptography. Our method is based on the formulation of cyclotomic matrices correspond to the diophantine system. The proposed cyclotomic asymmetric cryptosystem (CAC) utilizes the cyclotomic matrices, whose entries are cyclotomic numbers of order $2l^{2}$, $l$ be prime over a finite field $\mathbb{F}_{p}$ of $p$ elements. The method utilize cyclotomic matrices to design a one-way function. The ou…
Cyclotomic Matrices and Power Difference Sets
本文通过将分圆矩阵与Schur环基元关联揭示其代数结构，得出结构恒等式并重新解释经典结果，还为幂差集问题建立了分圆矩阵表达的谱和行列式条件。
Cyclotomic matrices related to Kloosterman sums over finite fields
利用有限域上特征和的算术性质，研究了包含Kloosterman和的分圆矩阵，证明了矩阵[K_p(1,i^2+j^2)]_{1≤i,j≤(p-1)/2}在p≥11时奇异的充要条件。
Products involving the real parts of Jacobi sums and related cyclotomic matrices
本文研究奇素数幂q下乘积R_q(χ_q)的算术性质，该乘积与Jacobi和的实部相关，并揭示了其与分圆矩阵的联系。
The Gauss periods and cyclotomic matrices involving Gauss sums over cyclic groups
利用高斯周期和特征和的性质，研究循环群上由高斯和定义的循环矩阵的行列式与特征值。
On cyclotomic matrices involving Gauss sums over finite fields
本文受前人分圆矩阵研究启发，研究有限域上涉及高斯和的分圆矩阵，给出了一类由高斯和构成的分圆矩阵的行列式的精确计算公式。
Hypothetical connection of the theta functions of Dirichlet characters with the real cyclotomic fields
本文提出一种基于theta函数泛函方程特殊形式的方法来研究Dirichlet L-函数的Lindelöf假设，通过求解线性方程组估计L函数值，数值实验揭示逆矩阵极限属于实分圆域。
The Pell sequence and cyclotomic matrices involving squares over finite fields
本文利用佩尔序列的算术性质和p进工具，研究有限域中涉及平方的环同态矩阵。证明当f≥2时，特定矩阵为奇异矩阵，并在q=p时，行列式为零当且仅当Q_p≡2 mod p²Z。
The problem of determining cyclotomic numbers in terms of the solutions of certain Diophantine systems has been treated by many authors since the age of Gauss. In this paper we obtain an explicit expression for cyclotomic numbers of order 2l^2 in terms of the coefficients of the Jacobi sums of lower orders. At the end, we illustrate the nature of two matrices corresponding to two types of cyclotomic numbers.
In this paper we study the non-existence problem of (nearly) perfect (almost) $m$-ary sequences via their connection to (near) Butson-Hadamard (BH) matrices and (near) conference matrices. Firstly, we apply a result on vanishing sums of roots of unity and a result of Brock on the unsolvability of certain equations over a cyclotomic number field to derive non-existence results for near BH matrices and near conference matrices. Secondly, we refine the idea of Brock in the case of cyclotomic number fields whose ring of integers is not a principal ideal domains and get many new non-existence resul…
Gaussian hypergeometric functions and cyclotomic matrices
基于高斯超几何函数与循环矩阵，研究非零平方元素的算术性质，推导出循环矩阵行列式的表达式，涉及雅可比和与高斯和的乘积形式。
For each odd prime $p$, let $ζ_p$ denote a primitive $p$-th root of unity. In this paper, we study the determinants of some matrices with cyclotomic unit entries. For instance, we show that when $p\equiv 3\pmod4$ and $p>3$ the determinant of the matrix $\(\frac{1-ζ_p^{j^2k^2}}{1-ζ_p^{j^2}}\)_{1\le j,k\le (p-1)/2}$ can be written as $(-1)^{\frac{h(-p)+1}{2}}(a_p+b_pi\sqrt{p})$ with $a_p,b_p\in\frac12\Z$ and $$\begin{cases}ν_p(a_p)=ν_p(b_p)=\frac{p-3}{8}&\mbox{if}\ p\equiv 3\pmod8, \\ν_p(a_p)=ν_p(b_p)+1=\frac{p+1}{8}&\mbox{if}\ p\equiv 7\pmod8,\end{cases}$$ where $ν_p(x)$ denotes the $p$-adic or…
For an integer $n\geq 8$ divisible by $4$, let $R_n=\mathbb{Z}[ζ_n,1/2]$ and let $\operatorname{U}_2(R_n)$ be the group of $2\times 2$ unitary matrices with entries in $R_n$. Set $\operatorname{U}_2^ζ(R_n)=\{γ\in\operatorname{U}_2(R_n)\mid \detγ\in\langleζ_n\rangle\}$. Let $\mathcal{G}_n\subseteq \operatorname{U}_2^ζ(R_n)$ be the Clifford-cyclotomic group generated by a Hadamard matrix $H=\frac{1}{2}[\begin{smallmatrix} 1+i & 1+i\\1+i &-1-i\end{smallmatrix}]$ and the gate $T=[\begin{smallmatrix}1 & 0\\0 & ζ_n\end{smallmatrix}]$. We prove that $\mathcal{G}_n=\operatorname{U}_2^ζ(R_n)$ if and on…
On finite field analogues of determinants involving the Beta function
本文研究有限域上的Jacobi和矩阵，推导出其行列式公式，类比Beta函数的行列式结果。
Hecke's inverse cotangent numbers
本文研究主同余子群 Eisenstein 级数中 Hecke 引入的数，证明其本质是分圆域或极大实子域的生成元，通过矩阵求逆由余切数得到，并给出了无平方因子下用单位根线性组合及广义 Bernoulli 数和 Gauss 和表示的公式。
Tribonacci properties of identities, matrices, and determinants
本文通过矩阵方法和二项式反演技术获得 Tribonacci 数的对称恒等式，并利用 Bell 多项式推导出 Tribonacci 数在修正的 Toeplitz-Hessenberg 形式下的行列式表示。
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